Theorem List for Metamath Proof Explorer - 44801-44900 *Has distinct variable
group(s)
| Type | Label | Description |
|---|
| Statement |
|---|
| |
| Theorem | snsslVD 44801 |
Virtual deduction proof of snssl 44802. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
| |
| Theorem | snssl 44802 |
If a singleton is a subclass of another class, then the singleton's
element is an element of that other class. This theorem is the
right-to-left implication of the biconditional snss 4761.
The proof of
this theorem was automatically generated from snsslVD 44801 using a tools
command file, translateMWO.cmd, by translating the proof into its
non-virtual deduction form and minimizing it. (Contributed by Alan
Sare, 25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
| |
| Theorem | snelpwrVD 44803 |
Virtual deduction proof of snelpwi 5418. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
| |
| Theorem | unipwrVD 44804 |
Virtual deduction proof of unipwr 44805. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
| |
| Theorem | unipwr 44805 |
A class is a subclass of the union of its power class. This theorem is
the right-to-left subclass lemma of unipw 5425. The proof of this theorem
was automatically generated from unipwrVD 44804 using a tools command file ,
translateMWO.cmd , by translating the proof into its non-virtual
deduction form and minimizing it. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
| |
| Theorem | sstrALT2VD 44806 |
Virtual deduction proof of sstrALT2 44807. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
| |
| Theorem | sstrALT2 44807 |
Virtual deduction proof of sstr 3967, transitivity of subclasses, Theorem
6 of [Suppes] p. 23. This theorem was
automatically generated from
sstrALT2VD 44806 using the command file
translate_without_overwriting.cmd . It was not minimized because the
automated minimization excluding duplicates generates a minimized proof
which, although not directly containing any duplicates, indirectly
contains a duplicate. That is, the trace back of the minimized proof
contains a duplicate. This is undesirable because some step(s) of the
minimized proof use the proven theorem. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
| |
| Theorem | suctrALT2VD 44808 |
Virtual deduction proof of suctrALT2 44809. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | suctrALT2 44809 |
Virtual deduction proof of suctr 6439. The successor of a transitive
class is transitive. This proof was generated automatically from the
virtual deduction proof suctrALT2VD 44808 using the tools command file
translate_without_overwriting_minimize_excluding_duplicates.cmd .
(Contributed by Alan Sare, 11-Sep-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | elex2VD 44810* |
Virtual deduction proof of elex2 2811. (Contributed by Alan Sare,
25-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
| |
| Theorem | elex22VD 44811* |
Virtual deduction proof of elex22 3485. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) |
| |
| Theorem | eqsbc2VD 44812* |
Virtual deduction proof of eqsbc2 3829. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴)) |
| |
| Theorem | zfregs2VD 44813* |
Virtual deduction proof of zfregs2 9745. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ≠ ∅ → ¬
∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) |
| |
| Theorem | tpid3gVD 44814 |
Virtual deduction proof of tpid3g 4748. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
| |
| Theorem | en3lplem1VD 44815* |
Virtual deduction proof of en3lplem1 9624. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))) |
| |
| Theorem | en3lplem2VD 44816* |
Virtual deduction proof of en3lplem2 9625. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))) |
| |
| Theorem | en3lpVD 44817 |
Virtual deduction proof of en3lp 9626. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) |
| |
| 21.41.7 Theorems proved using Virtual Deduction
with mmj2 assistance
|
| |
| Theorem | simplbi2VD 44818 |
Virtual deduction proof of simplbi2 500. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
| | 3:1,?: e0a 44744 | ⊢ ((𝜓 ∧ 𝜒) → 𝜑)
| | qed:3,?: e0a 44744 | ⊢ (𝜓 → (𝜒 → 𝜑))
|
The proof of simplbi2 500 was automatically derived from it.
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 → (𝜒 → 𝜑)) |
| |
| Theorem | 3ornot23VD 44819 |
Virtual deduction proof of 3ornot23 44482. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ (¬ 𝜑
∧ ¬ 𝜓) )
| | 2:: | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ (𝜒 ∨ 𝜑 ∨ 𝜓) )
| | 3:1,?: e1a 44600 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜑 )
| | 4:1,?: e1a 44600 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜓 )
| | 5:3,4,?: e11 44661 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ (𝜑
∨ 𝜓) )
| | 6:2,?: e2 44604 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ (𝜒 ∨ (𝜑 ∨ 𝜓)) )
| | 7:5,6,?: e12 44696 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ 𝜒 )
| | 8:7: | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ((𝜒
∨ 𝜑 ∨ 𝜓) → 𝜒) )
| | qed:8: | ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒
∨ 𝜑 ∨ 𝜓) → 𝜒))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒)) |
| |
| Theorem | orbi1rVD 44820 |
Virtual deduction proof of orbi1r 44483. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| | 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜒 ∨ 𝜑) )
| | 3:2,?: e2 44604 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜑 ∨ 𝜒) )
| | 4:1,3,?: e12 44696 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜓 ∨ 𝜒) )
| | 5:4,?: e2 44604 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜒 ∨ 𝜓) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜑)
→ (𝜒 ∨ 𝜓)) )
| | 7:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜓) )
| | 8:7,?: e2 44604 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜓 ∨ 𝜒) )
| | 9:1,8,?: e12 44696 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜑 ∨ 𝜒) )
| | 10:9,?: e2 44604 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜑) )
| | 11:10: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜓)
→ (𝜒 ∨ 𝜑)) )
| | 12:6,11,?: e11 44661 | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒
∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) )
| | qed:12: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑)
↔ (𝜒 ∨ 𝜓)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))) |
| |
| Theorem | bitr3VD 44821 |
Virtual deduction proof of bitr3 352. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑
↔ 𝜓) )
| | 2:1,?: e1a 44600 | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜓
↔ 𝜑) )
| | 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜑 ↔ 𝜒) )
| | 4:3,?: e2 44604 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜒 ↔ 𝜑) )
| | 5:2,4,?: e12 44696 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜓 ↔ 𝜒) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜑
↔ 𝜒) → (𝜓 ↔ 𝜒)) )
| | qed:6: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒)
→ (𝜓 ↔ 𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒))) |
| |
| Theorem | 3orbi123VD 44822 |
Virtual deduction proof of 3orbi123 44484. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧
(𝜏 ↔ 𝜂)) )
| | 2:1,?: e1a 44600 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜑 ↔ 𝜓) )
| | 3:1,?: e1a 44600 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜒 ↔ 𝜃) )
| | 4:1,?: e1a 44600 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜏 ↔ 𝜂) )
| | 5:2,3,?: e11 44661 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) )
| | 6:5,4,?: e11 44661 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 7:?: | ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ (𝜑
∨ 𝜒 ∨ 𝜏))
| | 8:6,7,?: e10 44667 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 9:?: | ⊢ (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔
(𝜓 ∨ 𝜃 ∨ 𝜂))
| | 10:8,9,?: e10 44667 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒
↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨
𝜃 ∨ 𝜂)) )
| | qed:10: | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃
∨ 𝜂)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))) |
| |
| Theorem | sbc3orgVD 44823 |
Virtual deduction proof of the analogue of sbcor 3816 with three disjuncts.
The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:1,?: e1a 44600 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝜑
∨ 𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 3:: | ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑
∨ 𝜓 ∨ 𝜒))
| | 32:3: | ⊢ ∀𝑥(((𝜑 ∨ 𝜓) ∨ 𝜒)
↔ (𝜑 ∨ 𝜓 ∨ 𝜒))
| | 33:1,32,?: e10 44667 | ⊢ ( 𝐴 ∈ 𝐵 ▶ [𝐴 / 𝑥](((𝜑
∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 4:1,33,?: e11 44661 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝜑
∨ 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 5:2,4,?: e11 44661 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒)) )
| | 6:1,?: e1a 44600 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) )
| | 7:6,?: e1a 44600 | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥](𝜑
∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 8:5,7,?: e11 44661 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 9:?: | ⊢ ((([𝐴 / 𝑥]𝜑
∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑
∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))
| | 10:8,9,?: e10 44667 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓
∨ [𝐴 / 𝑥]𝜒)) )
| | qed:10: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓
∨ [𝐴 / 𝑥]𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))) |
| |
| Theorem | 19.21a3con13vVD 44824* |
Virtual deduction proof of alrim3con13v 44506. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 → ∀𝑥𝜑)
▶ (𝜑 → ∀𝑥𝜑) )
| | 2:: | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓 ∧ 𝜑
∧ 𝜒) ▶ (𝜓 ∧ 𝜑 ∧ 𝜒) )
| | 3:2,?: e2 44604 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜓 )
| | 4:2,?: e2 44604 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜑 )
| | 5:2,?: e2 44604 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜒 )
| | 6:1,4,?: e12 44696 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜑 )
| | 7:3,?: e2 44604 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜓 )
| | 8:5,?: e2 44604 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜒 )
| | 9:7,6,8,?: e222 44609 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒) )
| | 10:9,?: e2 44604 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒) )
| | 11:10:in2 | ⊢ ( (𝜑 → ∀𝑥𝜑) ▶ ((𝜓
∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)) )
| | qed:11:in1 | ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓
∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒))) |
| |
| Theorem | exbirVD 44825 |
Virtual deduction proof of exbir 44452. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
▶ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) )
| | 2:: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ,
(𝜑 ∧ 𝜓) ▶ (𝜑 ∧ 𝜓) )
| | 3:: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ,
(𝜑 ∧ 𝜓), 𝜃 ▶ 𝜃 )
| | 5:1,2,?: e12 44696 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓) ▶ (𝜒 ↔ 𝜃) )
| | 6:3,5,?: e32 44730 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓), 𝜃 ▶ 𝜒 )
| | 7:6: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓) ▶ (𝜃 → 𝜒) )
| | 8:7: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
▶ ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒)) )
| | 9:8,?: e1a 44600 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)) ▶ (𝜑 → (𝜓 → (𝜃 → 𝜒))) )
| | qed:9: | ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
→ (𝜑 → (𝜓 → (𝜃 → 𝜒))))
|
(Contributed by Alan Sare, 13-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) |
| |
| Theorem | exbiriVD 44826 |
Virtual deduction proof of exbiri 810. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
| | 2:: | ⊢ ( 𝜑 ▶ 𝜑 )
| | 3:: | ⊢ ( 𝜑 , 𝜓 ▶ 𝜓 )
| | 4:: | ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜃 )
| | 5:2,1,?: e10 44667 | ⊢ ( 𝜑 ▶ (𝜓 → (𝜒 ↔ 𝜃)) )
| | 6:3,5,?: e21 44702 | ⊢ ( 𝜑 , 𝜓 ▶ (𝜒 ↔ 𝜃) )
| | 7:4,6,?: e32 44730 | ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜒 )
| | 8:7: | ⊢ ( 𝜑 , 𝜓 ▶ (𝜃 → 𝜒) )
| | 9:8: | ⊢ ( 𝜑 ▶ (𝜓 → (𝜃 → 𝜒)) )
| | qed:9: | ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) |
| |
| Theorem | rspsbc2VD 44827* |
Virtual deduction proof of rspsbc2 44507. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:: | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ 𝐶 ∈ 𝐷 )
| | 3:: | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 )
| | 4:1,3,?: e13 44720 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑 )
| | 5:1,4,?: e13 44720 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑 )
| | 6:2,5,?: e23 44727 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑 )
| | 7:6: | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ (∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑) )
| | 8:7: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝐶 ∈ 𝐷
→ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) )
| | qed:8: | ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷
→ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))) |
| |
| Theorem | 3impexpVD 44828 |
Virtual deduction proof of 3impexp 1359. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) )
| | 2:: | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒)
↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
| | 3:1,2,?: e10 44667 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 4:3,?: e1a 44600 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 5:4,?: e1a 44600 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ (𝜑 → (𝜓 → (𝜒 → 𝜃))) )
| | 6:5: | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
→ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
| | 7:: | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ (𝜑 → (𝜓 → (𝜒 → 𝜃))) )
| | 8:7,?: e1a 44600 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 9:8,?: e1a 44600 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 10:2,9,?: e01 44664 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) )
| | 11:10: | ⊢ ((𝜑 → (𝜓 → (𝜒
→ 𝜃))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
| | qed:6,11,?: e00 44740 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃)))) |
| |
| Theorem | 3impexpbicomVD 44829 |
Virtual deduction proof of 3impexpbicom 44453. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) )
| | 2:: | ⊢ ((𝜃 ↔ 𝜏) ↔ (𝜏
↔ 𝜃))
| | 3:1,2,?: e10 44667 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃)) )
| | 4:3,?: e1a 44600 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))) )
| | 5:4: | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) → (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))))
| | 6:: | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))) )
| | 7:6,?: e1a 44600 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏
↔ 𝜃)) )
| | 8:7,2,?: e10 44667 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃
↔ 𝜏)) )
| | 9:8: | ⊢ ((𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃
↔ 𝜏)))
| | qed:5,9,?: e00 44740 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))) |
| |
| Theorem | 3impexpbicomiVD 44830 |
Virtual deduction proof of 3impexpbicomi 44454. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃
↔ 𝜏))
| | qed:1,?: e0a 44744 | ⊢ (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃))))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))) |
| |
| Theorem | sbcoreleleqVD 44831* |
Virtual deduction proof of sbcoreleleq 44508. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:1,?: e1a 44600 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 ∈
𝑦 ↔ 𝑥 ∈ 𝐴) )
| | 3:1,?: e1a 44600 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑦 ∈
𝑥 ↔ 𝐴 ∈ 𝑥) )
| | 4:1,?: e1a 44600 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 =
𝑦 ↔ 𝑥 = 𝐴) )
| | 5:2,3,4,?: e111 44647 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ((𝑥 ∈ 𝐴
∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥
∨ [𝐴 / 𝑦]𝑥 = 𝑦)) )
| | 6:1,?: e1a 44600 | ⊢ ( 𝐴 ∈ 𝐵
▶ ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥
∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥 ∨ [𝐴 / 𝑦]𝑥 = 𝑦)) )
| | 7:5,6: e11 44661 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦](𝑥
∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)) )
| | qed:7: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴))) |
| |
| Theorem | hbra2VD 44832* |
Virtual deduction proof of nfra2 3355. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ (∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑 →
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 2:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 3:1,2,?: e00 44740 | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 →
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 4:2: | ⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 5:4,?: e0a 44744 | ⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | qed:3,5,?: e00 44740 | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 →
∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑)
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) |
| |
| Theorem | tratrbVD 44833* |
Virtual deduction proof of tratrb 44509. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)
∧ 𝐵 ∈ 𝐴) )
| | 2:1,?: e1a 44600 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| | 3:1,?: e1a 44600 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 4:1,?: e1a 44600 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ∈ 𝐴 )
| | 5:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) )
| | 6:5,?: e2 44604 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝑦 )
| | 7:5,?: e2 44604 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑦 ∈ 𝐵 )
| | 8:2,7,4,?: e121 44629 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑦 ∈ 𝐴 )
| | 9:2,6,8,?: e122 44626 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝐴 )
| | 10:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥 ▶ 𝐵 ∈ 𝑥 )
| | 11:6,7,10,?: e223 44608 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥 ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥) )
| | 12:11: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝐵 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥)) )
| | 13:: | ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵
∧ 𝐵 ∈ 𝑥)
| | 14:12,13,?: e20 44699 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝐵 ∈ 𝑥 )
| | 15:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ 𝑥 = 𝐵 )
| | 16:7,15,?: e23 44727 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ 𝑦 ∈ 𝑥 )
| | 17:6,16,?: e23 44727 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) )
| | 18:17: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 = 𝐵 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)) )
| | 19:: | ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)
| | 20:18,19,?: e20 44699 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝑥 = 𝐵 )
| | 21:3,?: e1a 44600 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑦 ∈ 𝐴
∀𝑥 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 22:21,9,4,?: e121 44629 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦) )
| | 23:22,?: e2 44604 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ [𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 24:4,23,?: e12 44696 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵) )
| | 25:14,20,24,?: e222 44609 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝐵 )
| | 26:25: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ((𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| | 27:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨
𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 28:27,?: e0a 44744 | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
→ ∀𝑦(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
| | 29:28,26: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| | 30:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑥∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 31:30,?: e0a 44744 | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑥(Tr 𝐴
∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
| | 32:31,29: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥
∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| | 33:32,?: e1a 44600 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐵 )
| | qed:33: | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵) |
| |
| Theorem | al2imVD 44834 |
Virtual deduction proof of al2im 1814. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ ∀𝑥(𝜑 → (𝜓 → 𝜒)) )
| | 2:1,?: e1a 44600 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)) )
| | 3:: | ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓
→ ∀𝑥𝜒))
| | 4:2,3,?: e10 44667 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) )
| | qed:4: | ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒))
→ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))) |
| |
| Theorem | syl5impVD 44835 |
Virtual deduction proof of syl5imp 44485. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜑
→ (𝜓 → 𝜒)) )
| | 2:1,?: e1a 44600 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜓
→ (𝜑 → 𝜒)) )
| | 3:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → 𝜓) )
| | 4:3,2,?: e21 44702 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → (𝜑 → 𝜒)) )
| | 5:4,?: e2 44604 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜑 → (𝜃 → 𝜒)) )
| | 6:5: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ ((𝜃
→ 𝜓) → (𝜑 → (𝜃 → 𝜒))) )
| | qed:6: | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃
→ 𝜓) → (𝜑 → (𝜃 → 𝜒))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒)))) |
| |
| Theorem | idiVD 44836 |
Virtual deduction proof of idiALT 44451. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ 𝜑
| | qed:1,?: e0a 44744 | ⊢ 𝜑
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝜑 ⇒ ⊢ 𝜑 |
| |
| Theorem | ancomstVD 44837 |
Closed form of ancoms 458. The following user's proof is completed by
invoking mmj2's unify command and using mmj2's StepSelector to pick all
remaining steps of the Metamath proof.
| 1:: | ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
| | qed:1,?: e0a 44744 | ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓
∧ 𝜑) → 𝜒))
|
The proof of ancomst 464 is derived automatically from it.
(Contributed by
Alan Sare, 25-Dec-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒)) |
| |
| Theorem | ssralv2VD 44838* |
Quantification restricted to a subclass for two quantifiers. ssralv 4027
for two quantifiers. The following User's Proof is a Virtual Deduction
proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. ssralv2 44504 is ssralv2VD 44838 without
virtual deductions and was automatically derived from ssralv2VD 44838.
| 1:: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ▶ (𝐴 ⊆ 𝐵
∧ 𝐶 ⊆ 𝐷) )
| | 2:: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 )
| | 3:1: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ▶ 𝐴 ⊆ 𝐵 )
| | 4:3,2: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐷𝜑 )
| | 5:4: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑) )
| | 6:5: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑) )
| | 7:: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 )
| | 8:7,6: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴 ▶ ∀𝑦 ∈ 𝐷𝜑 )
| | 9:1: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ▶ 𝐶 ⊆ 𝐷 )
| | 10:9,8: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴 ▶ ∀𝑦 ∈ 𝐶𝜑 )
| | 11:10: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑) )
| | 12:: | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
→ ∀𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷))
| | 13:: | ⊢ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑
→ ∀𝑥∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑)
| | 14:12,13,11: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑) )
| | 15:14: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑 )
| | 16:15: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
▶ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑) )
| | qed:16: | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
→ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑))
|
(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)) |
| |
| Theorem | ordelordALTVD 44839 |
An element of an ordinal class is ordinal. Proposition 7.6 of
[TakeutiZaring] p. 36. This is an alternate proof of ordelord 6374 using
the Axiom of Regularity indirectly through dford2 9632. dford2 is a
weaker definition of ordinal number. Given the Axiom of Regularity, it
need not be assumed that E Fr 𝐴 because this is inferred by the
Axiom of Regularity. The following User's Proof is a Virtual Deduction
proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. ordelordALT 44510 is ordelordALTVD 44839
without virtual deductions and was automatically derived from
ordelordALTVD 44839 using the tools program
translate..without..overwriting.cmd and the Metamath program "MM-PA>
MINIMIZE_WITH *" command.
| 1:: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ (Ord 𝐴
∧ 𝐵 ∈ 𝐴) )
| | 2:1: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Ord 𝐴 )
| | 3:1: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ∈ 𝐴 )
| | 4:2: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| | 5:2: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) )
| | 6:4,3: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ⊆ 𝐴 )
| | 7:6,6,5: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐵(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) )
| | 8:: | ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)
↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 9:8: | ⊢ ∀𝑦((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)
↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 10:9: | ⊢ ∀𝑦 ∈ 𝐴((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦
∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 11:10: | ⊢ (∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦
∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 12:11: | ⊢ ∀𝑥(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦
∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 13:12: | ⊢ ∀𝑥 ∈ 𝐴(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 14:13: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦))
| | 15:14,5: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 16:4,15,3: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐵 )
| | 17:16,7: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Ord 𝐵 )
| | qed:17: | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
|
(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) |
| |
| Theorem | equncomVD 44840 |
If a class equals the union of two other classes, then it equals the union
of those two classes commuted. The following User's Proof is a Virtual
Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. equncom 4134 is equncomVD 44840 without
virtual deductions and was automatically derived from equncomVD 44840.
| 1:: | ⊢ ( 𝐴 = (𝐵 ∪ 𝐶) ▶ 𝐴 = (𝐵 ∪ 𝐶) )
| | 2:: | ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
| | 3:1,2: | ⊢ ( 𝐴 = (𝐵 ∪ 𝐶) ▶ 𝐴 = (𝐶 ∪ 𝐵) )
| | 4:3: | ⊢ (𝐴 = (𝐵 ∪ 𝐶) → 𝐴 = (𝐶 ∪ 𝐵))
| | 5:: | ⊢ ( 𝐴 = (𝐶 ∪ 𝐵) ▶ 𝐴 = (𝐶 ∪ 𝐵) )
| | 6:5,2: | ⊢ ( 𝐴 = (𝐶 ∪ 𝐵) ▶ 𝐴 = (𝐵 ∪ 𝐶) )
| | 7:6: | ⊢ (𝐴 = (𝐶 ∪ 𝐵) → 𝐴 = (𝐵 ∪ 𝐶))
| | 8:4,7: | ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))
|
(Contributed by Alan Sare, 17-Feb-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) |
| |
| Theorem | equncomiVD 44841 |
Inference form of equncom 4134. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. equncomi 4135 is equncomiVD 44841 without
virtual deductions and was automatically derived from equncomiVD 44841.
| h1:: | ⊢ 𝐴 = (𝐵 ∪ 𝐶)
| | qed:1: | ⊢ 𝐴 = (𝐶 ∪ 𝐵)
|
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐴 = (𝐶 ∪ 𝐵) |
| |
| Theorem | sucidALTVD 44842 |
A set belongs to its successor. Alternate proof of sucid 6435.
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. sucidALT 44843 is sucidALTVD 44842
without virtual deductions and was automatically derived from
sucidALTVD 44842. This proof illustrates that
completeusersproof.cmd will generate a Metamath proof from any
User's Proof which is "conventional" in the sense that no step
is a virtual deduction, provided that all necessary unification
theorems and transformation deductions are in set.mm.
completeusersproof.cmd automatically converts such a
conventional proof into a Virtual Deduction proof for which each
step happens to be a 0-virtual hypothesis virtual deduction.
The user does not need to search for reference theorem labels or
deduction labels nor does he(she) need to use theorems and
deductions which unify with reference theorems and deductions in
set.mm. All that is necessary is that each theorem or deduction
of the User's Proof unifies with some reference theorem or
deduction in set.mm or is a semantic variation of some theorem
or deduction which unifies with some reference theorem or
deduction in set.mm. The definition of "semantic variation" has
not been precisely defined. If it is obvious that a theorem or
deduction has the same meaning as another theorem or deduction,
then it is a semantic variation of the latter theorem or
deduction. For example, step 4 of the User's Proof is a
semantic variation of the definition (axiom)
suc 𝐴 = (𝐴 ∪ {𝐴}), which unifies with df-suc 6358, a
reference definition (axiom) in set.mm. Also, a theorem or
deduction is said to be a semantic variation of another
theorem or deduction if it is obvious upon cursory inspection
that it has the same meaning as a weaker form of the latter
theorem or deduction. For example, the deduction Ord 𝐴
infers ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) is a
semantic variation of the theorem (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))), which unifies with
the set.mm reference definition (axiom) dford2 9632.
| h1:: | ⊢ 𝐴 ∈ V
| | 2:1: | ⊢ 𝐴 ∈ {𝐴}
| | 3:2: | ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴)
| | 4:: | ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴)
| | qed:3,4: | ⊢ 𝐴 ∈ suc 𝐴
|
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ 𝐴 ∈ suc 𝐴 |
| |
| Theorem | sucidALT 44843 |
A set belongs to its successor. This proof was automatically derived
from sucidALTVD 44842 using translate_without_overwriting.cmd and
minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ 𝐴 ∈ suc 𝐴 |
| |
| Theorem | sucidVD 44844 |
A set belongs to its successor. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools
program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant.
sucid 6435 is sucidVD 44844 without virtual deductions and was automatically
derived from sucidVD 44844.
| h1:: | ⊢ 𝐴 ∈ V
| | 2:1: | ⊢ 𝐴 ∈ {𝐴}
| | 3:2: | ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴})
| | 4:: | ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
| | qed:3,4: | ⊢ 𝐴 ∈ suc 𝐴
|
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ 𝐴 ∈ suc 𝐴 |
| |
| Theorem | imbi12VD 44845 |
Implication form of imbi12i 350. The following User's Proof is a Virtual
Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. imbi12 346 is imbi12VD 44845 without virtual
deductions and was automatically derived from imbi12VD 44845.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| | 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ (𝜒 ↔ 𝜃) )
| | 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜑
→ 𝜒) ▶ (𝜑 → 𝜒) )
| | 4:1,3: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜑
→ 𝜒) ▶ (𝜓 → 𝜒) )
| | 5:2,4: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜑
→ 𝜒) ▶ (𝜓 → 𝜃) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜑 → 𝜒) → (𝜓 → 𝜃)) )
| | 7:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜓
→ 𝜃) ▶ (𝜓 → 𝜃) )
| | 8:1,7: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜓
→ 𝜃) ▶ (𝜑 → 𝜃) )
| | 9:2,8: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜓
→ 𝜃) ▶ (𝜑 → 𝜒) )
| | 10:9: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜓 → 𝜃) → (𝜑 → 𝜒)) )
| | 11:6,10: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)) )
| | 12:11: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ↔ 𝜃)
→ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) )
| | qed:12: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃)
→ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))
|
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))) |
| |
| Theorem | imbi13VD 44846 |
Join three logical equivalences to form equivalence of implications. The
following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi13 44493
is imbi13VD 44846 without virtual deductions and was automatically derived
from imbi13VD 44846.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| | 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ (𝜒 ↔ 𝜃) )
| | 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ (𝜏 ↔ 𝜂) )
| | 4:2,3: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂)) )
| | 5:1,4: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂)))) )
| | 7:6: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ↔ 𝜃)
→ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂))))) )
| | qed:7: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃)
→ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂))))))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))) |
| |
| Theorem | sbcim2gVD 44847 |
Distribution of class substitution over a left-nested implication.
Similar to sbcimg 3814.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
sbcim2g 44511 is sbcim2gVD 44847 without virtual deductions and was automatically
derived from sbcim2gVD 44847.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:: | ⊢ ( 𝐴 ∈ 𝐵 , [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ▶ [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) )
| | 3:1,2: | ⊢ ( 𝐴 ∈ 𝐵 , [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ▶ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) )
| | 4:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜓 → 𝜒)
↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) )
| | 5:3,4: | ⊢ ( 𝐴 ∈ 𝐵 , [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ▶ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒)) )
| | 6:5: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒))) )
| | 7:: | ⊢ ( 𝐴 ∈ 𝐵 , ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) ▶ ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) )
| | 8:4,7: | ⊢ ( 𝐴 ∈ 𝐵 , ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) ▶ ([𝐴 / 𝑥]𝜑
→ [𝐴 / 𝑥](𝜓 → 𝜒)) )
| | 9:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))) )
| | 10:8,9: | ⊢ ( 𝐴 ∈ 𝐵 , ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) ▶ [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) )
| | 11:10: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒))) )
| | 12:6,11: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
→ (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒))) )
| | qed:12: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒))))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) |
| |
| Theorem | sbcbiVD 44848 |
Implication form of sbcbii 3822.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
sbcbi 44512 is sbcbiVD 44848 without virtual deductions and was automatically
derived from sbcbiVD 44848.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:: | ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓)
▶ ∀𝑥(𝜑 ↔ 𝜓) )
| | 3:1,2: | ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓)
▶ [𝐴 / 𝑥](𝜑 ↔ 𝜓) )
| | 4:1,3: | ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓)
▶ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) )
| | 5:4: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑥(𝜑 ↔ 𝜓)
→ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) )
| | qed:5: | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓)
→ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
| |
| Theorem | trsbcVD 44849* |
Formula-building inference rule for class substitution, substituting a
class variable for the setvar variable of the transitivity predicate.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
trsbc 44513 is trsbcVD 44849 without virtual deductions and was automatically
derived from trsbcVD 44849.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑧 ∈ 𝑦
↔ 𝑧 ∈ 𝑦) )
| | 3:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝑥
↔ 𝑦 ∈ 𝐴) )
| | 4:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑧 ∈ 𝑥
↔ 𝑧 ∈ 𝐴) )
| | 5:1,2,3,4: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝑧 ∈ 𝑦
→ ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))) )
| | 6:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 →
([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥))) )
| | 7:5,6: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴
→ 𝑧 ∈ 𝐴))) )
| | 8:: | ⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴
→ 𝑧 ∈ 𝐴)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
| | 9:7,8: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 10:: | ⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥
→ 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
| | 11:10: | ⊢ ∀𝑥((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥
→ 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
| | 12:1,11: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)
→ 𝑧 ∈ 𝑥)) )
| | 13:9,12: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 14:13: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑦([𝐴 / 𝑥]((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 15:14: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑦[𝐴 / 𝑥]((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 16:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) )
| | 17:15,16: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 18:17: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑧([𝐴 / 𝑥]∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 19:18: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑧[𝐴 / 𝑥]∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) )
| | 20:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑧∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) )
| | 21:19,20: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑧∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) )
| | 22:: | ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
| | 23:21,22: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑧∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴) )
| | 24:: | ⊢ (Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦
∈ 𝑥) → 𝑧 ∈ 𝑥))
| | 25:24: | ⊢ ∀𝑥(Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
| | 26:1,25: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]Tr 𝑥
↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) )
| | 27:23,26: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]Tr 𝑥
↔ Tr 𝐴) )
| | qed:27: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥
↔ Tr 𝐴))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴)) |
| |
| Theorem | truniALTVD 44850* |
The union of a class of transitive sets is transitive.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
truniALT 44514 is truniALTVD 44850 without virtual deductions and was
automatically derived from truniALTVD 44850.
| 1:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ∀𝑥 ∈ 𝐴
Tr 𝑥 )
| | 2:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) )
| | 3:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ 𝑧 ∈ 𝑦 )
| | 4:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ 𝑦 ∈ ∪ 𝐴 )
| | 5:4: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) )
| | 6:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) )
| | 7:6: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑦 ∈ 𝑞 )
| | 8:6: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑞 ∈ 𝐴 )
| | 9:1,8: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ [𝑞 / 𝑥]Tr 𝑥 )
| | 10:8,9: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ Tr 𝑞 )
| | 11:3,7,10: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑧 ∈ 𝑞 )
| | 12:11,8: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑧 ∈ ∪ 𝐴 )
| | 13:12: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 14:13: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 15:14: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 16:5,15: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ 𝑧 ∈ ∪ 𝐴 )
| | 17:16: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ((𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 18:17: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥
▶ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 19:18: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ Tr ∪ 𝐴 )
| | qed:19: | ⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∪ 𝐴)
|
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪
𝐴) |
| |
| Theorem | ee33VD 44851 |
Non-virtual deduction form of e33 44706.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
ee33 44494 is ee33VD 44851 without virtual deductions and was automatically
derived from ee33VD 44851.
| h1:: | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
| | h2:: | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
| | h3:: | ⊢ (𝜃 → (𝜏 → 𝜂))
| | 4:1,3: | ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
| | 5:4: | ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
| | 6:2,5: | ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓
→ (𝜒 → 𝜂))))))
| | 7:6: | ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒
→ 𝜂)))))
| | 8:7: | ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
| | qed:8: | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
|
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) |
| |
| Theorem | trintALTVD 44852* |
The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 44853.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
trintALT 44853 is trintALTVD 44852 without virtual deductions and was
automatically derived from trintALTVD 44852.
| 1:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ∀𝑥 ∈ 𝐴Tr 𝑥 )
| | 2:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) )
| | 3:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ 𝑧 ∈ 𝑦 )
| | 4:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ 𝑦 ∈ ∩ 𝐴 )
| | 5:4: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ ∀𝑞 ∈ 𝐴𝑦 ∈ 𝑞 )
| | 6:5: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞) )
| | 7:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ 𝑞 ∈ 𝐴 )
| | 8:7,6: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ 𝑦 ∈ 𝑞 )
| | 9:7,1: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ [𝑞 / 𝑥]Tr 𝑥 )
| | 10:7,9: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ Tr 𝑞 )
| | 11:10,3,8: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ 𝑧 ∈ 𝑞 )
| | 12:11: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞) )
| | 13:12: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞) )
| | 14:13: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ ∀𝑞 ∈ 𝐴𝑧 ∈ 𝑞 )
| | 15:3,14: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ 𝑧 ∈ ∩ 𝐴 )
| | 16:15: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ((𝑧 ∈ 𝑦 ∧ 𝑦
∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴) )
| | 17:16: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ∀𝑧∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴) )
| | 18:17: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ Tr ∩ 𝐴 )
| | qed:18: | ⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∩ 𝐴)
|
(Contributed by Alan Sare, 17-Apr-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩
𝐴) |
| |
| Theorem | trintALT 44853* |
The intersection of a class of transitive sets is transitive. Exercise
5(b) of [Enderton] p. 73. trintALT 44853 is an alternate proof of trint 5247.
trintALT 44853 is trintALTVD 44852 without virtual deductions and was
automatically derived from trintALTVD 44852 using the tools program
translate..without..overwriting.cmd and the Metamath program
"MM-PA>
MINIMIZE_WITH *" command. (Contributed by Alan Sare, 17-Apr-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩
𝐴) |
| |
| Theorem | undif3VD 44854 |
The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual
deduction proof of undif3 4275.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
undif3 4275 is undif3VD 44854 without virtual deductions and was automatically
derived from undif3VD 44854.
| 1:: | ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴
∨ 𝑥 ∈ (𝐵 ∖ 𝐶)))
| | 2:: | ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈
𝐶))
| | 3:2: | ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥
∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 4:1,3: | ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴
∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 5:: | ⊢ ( 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 )
| | 6:5: | ⊢ ( 𝑥 ∈ 𝐴 ▶ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) )
| | 7:5: | ⊢ ( 𝑥 ∈ 𝐴 ▶ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴) )
| | 8:6,7: | ⊢ ( 𝑥 ∈ 𝐴 ▶ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧
(¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) )
| | 9:8: | ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (
¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| | 10:: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐵
∧ ¬ 𝑥 ∈ 𝐶) )
| | 11:10: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ 𝑥 ∈ 𝐵 )
| | 12:10: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ ¬ 𝑥 ∈ 𝐶
)
| | 13:11: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐴
∨ 𝑥 ∈ 𝐵) )
| | 14:12: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (¬ 𝑥 ∈
𝐶 ∨ 𝑥 ∈ 𝐴) )
| | 15:13,14: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ ((𝑥 ∈
𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) )
| | 16:15: | ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴
∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| | 17:9,16: | ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
→ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| | 18:: | ⊢ ( (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐴
∧ ¬ 𝑥 ∈ 𝐶) )
| | 19:18: | ⊢ ( (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ▶ 𝑥 ∈ 𝐴 )
| | 20:18: | ⊢ ( (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ▶ ¬ 𝑥 ∈ 𝐶
)
| | 21:18: | ⊢ ( (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐴
∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) )
| | 22:21: | ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨
(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 23:: | ⊢ ( (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ▶ (𝑥 ∈ 𝐴 ∧
𝑥 ∈ 𝐴) )
| | 24:23: | ⊢ ( (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ▶ 𝑥 ∈ 𝐴 )
| | 25:24: | ⊢ ( (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ▶ (𝑥 ∈ 𝐴 ∨
(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) )
| | 26:25: | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ (
𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 27:10: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐴
∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) )
| | 28:27: | ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨
(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 29:: | ⊢ ( (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ▶ (𝑥 ∈ 𝐵 ∧
𝑥 ∈ 𝐴) )
| | 30:29: | ⊢ ( (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ▶ 𝑥 ∈ 𝐴 )
| | 31:30: | ⊢ ( (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ▶ (𝑥 ∈ 𝐴 ∨
(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) )
| | 32:31: | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ (
𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 33:22,26: | ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐴
∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 34:28,32: | ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵
∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 35:33,34: | ⊢ ((((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈
𝐴 ∧ 𝑥 ∈ 𝐴)) ∨ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
→ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 36:: | ⊢ ((((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈
𝐴 ∧ 𝑥 ∈ 𝐴)) ∨ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| | 37:36,35: | ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶
∨ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| | 38:17,37: | ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| | 39:: | ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈
𝐴))
| | 40:39: | ⊢ (¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐶 ∧
¬ 𝑥 ∈ 𝐴))
| | 41:: | ⊢ (¬ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴) ↔ (¬ 𝑥
∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
| | 42:40,41: | ⊢ (¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥
∈ 𝐴))
| | 43:: | ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵
))
| | 44:43,42: | ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴)
) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴)))
| | 45:: | ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ (
𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴)))
| | 46:45,44: | ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ (
(𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| | 47:4,38: | ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴
∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| | 48:46,47: | ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴
∪ 𝐵) ∖ (𝐶 ∖ 𝐴)))
| | 49:48: | ⊢ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈
((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)))
| | qed:49: | ⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶
∖ 𝐴))
|
(Contributed by Alan Sare, 17-Apr-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) |
| |
| Theorem | sbcssgVD 44855 |
Virtual deduction proof of sbcssg 4495.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
sbcssg 4495 is sbcssgVD 44855 without virtual deductions and was automatically
derived from sbcssgVD 44855.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐶) )
| | 3:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐷) )
| | 4:2,3: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 →
[𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷
)) )
| | 5:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑦 ∈ 𝐶 →
𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷)) )
| | 6:4,5: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑦 ∈ 𝐶 →
𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| | 7:6: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑦([𝐴 / 𝑥](𝑦 ∈
𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| | 8:7: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑦[𝐴 / 𝑥](𝑦 ∈
𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)
) )
| | 9:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈
𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) )
| | 10:8,9: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈
𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)
) )
| | 11:: | ⊢ (𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
| | 110:11: | ⊢ ∀𝑥(𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈
𝐷))
| | 12:1,110: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔
[𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) )
| | 13:10,12: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔
∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| | 14:: | ⊢ (⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ ∀
𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))
| | 15:13,14: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔
⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷) )
| | qed:15: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋
𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
|
(Contributed by Alan Sare, 22-Jul-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷)) |
| |
| Theorem | csbingVD 44856 |
Virtual deduction proof of csbin 4417.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbin 4417 is csbingVD 44856 without virtual deductions and was
automatically derived from csbingVD 44856.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:: | ⊢ (𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)
}
| | 20:2: | ⊢ ∀𝑥(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦
∈ 𝐷)}
| | 30:1,20: | ⊢ ( 𝐴 ∈ 𝐵 ▶ [𝐴 / 𝑥](𝐶 ∩ 𝐷) =
{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} )
| | 3:1,30: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) =
⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} )
| | 4:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶
∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} )
| | 5:3,4: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) =
{𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} )
| | 6:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐶) )
| | 7:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐷) )
| | 8:6,7: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧
[𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷
)) )
| | 9:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧
𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷)) )
| | 10:9,8: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧
𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| | 11:10: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑦([𝐴 / 𝑥](𝑦 ∈
𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| | 12:11: | ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶
∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} )
| | 13:5,12: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) =
{𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} )
| | 14:: | ⊢ (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) = {
𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}
| | 15:13,14: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) =
(⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) )
| | qed:15: | ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (
⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)) |
| |
| Theorem | onfrALTlem5VD 44857* |
Virtual deduction proof of onfrALTlem5 44515.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem5 44515 is onfrALTlem5VD 44857 without virtual deductions and was
automatically derived from onfrALTlem5VD 44857.
| 1:: | ⊢ 𝑎 ∈ V
| | 2:1: | ⊢ (𝑎 ∩ 𝑥) ∈ V
| | 3:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ (𝑎
∩ 𝑥) = ∅)
| | 4:3: | ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔
¬ (𝑎 ∩ 𝑥) = ∅)
| | 5:: | ⊢ ((𝑎 ∩ 𝑥) ≠ ∅ ↔ ¬ (𝑎 ∩ 𝑥
) = ∅)
| | 6:4,5: | ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔
(𝑎 ∩ 𝑥) ≠ ∅)
| | 7:2: | ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔
[(𝑎 ∩ 𝑥) / 𝑏]¬ 𝑏 = ∅)
| | 8:: | ⊢ (𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅)
| | 9:8: | ⊢ ∀𝑏(𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅)
| | 10:2,9: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅ ↔
[(𝑎 ∩ 𝑥) / 𝑏]¬ 𝑏 = ∅)
| | 11:7,10: | ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔
[(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅)
| | 12:6,11: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅ ↔ (
𝑎 ∩ 𝑥) ≠ ∅)
| | 13:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥
) ↔ (𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥))
| | 14:12,13: | ⊢ (([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩
𝑥) ∧ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅) ↔ ((𝑎 ∩ 𝑥) ⊆ (𝑎
∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅))
| | 15:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩
𝑥) ∧ 𝑏 ≠ ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥) ∧
[(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅))
| | 16:15,14: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩
𝑥) ∧ 𝑏 ≠ ∅) ↔ ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥)
≠ ∅))
| | 17:2: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = (
⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 ∩ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦)
| | 18:2: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 = (𝑎 ∩ 𝑥)
| | 19:2: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦 = 𝑦
| | 20:18,19: | ⊢ (⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 ∩ ⦋(𝑎
∩ 𝑥) / 𝑏⦌𝑦) = ((𝑎 ∩ 𝑥) ∩ 𝑦)
| | 21:17,20: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ((
𝑎 ∩ 𝑥) ∩ 𝑦)
| | 22:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) =
∅ ↔ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ⦋(𝑎 ∩ 𝑥) / 𝑏⦌
∅)
| | 23:2: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌∅ = ∅
| | 24:21,23: | ⊢ (⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) =
⦋(𝑎 ∩ 𝑥) / 𝑏⦌∅ ↔ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
| | 25:22,24: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) =
∅ ↔ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
| | 26:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ↔ 𝑦 ∈
(𝑎 ∩ 𝑥))
| | 27:25,26: | ⊢ (([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ∧ [
(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((
𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
| | 28:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏
∩ 𝑦) = ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ∧ [(𝑎 ∩ 𝑥)
/ 𝑏](𝑏 ∩ 𝑦) = ∅))
| | 29:27,28: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏
∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) =
∅))
| | 30:29: | ⊢ ∀𝑦([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅))
| | 31:30: | ⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥)
∩ 𝑦) = ∅))
| | 32:: | ⊢ (∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩
𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅
))
| | 33:31,32: | ⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦)
= ∅)
| | 34:2: | ⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ (
𝑏 ∩ 𝑦) = ∅))
| | 35:33,34: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅)
| | 36:: | ⊢ (∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ ∃𝑦
(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅))
| | 37:36: | ⊢ ∀𝑏(∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔
∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅))
| | 38:2,37: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏
∩ 𝑦) = ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦)
= ∅))
| | 39:35,38: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏
∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
| | 40:16,39: | ⊢ (([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩
𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠
∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
| | 41:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ ([(𝑎
∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎 ∩ 𝑥) /
𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅))
| | qed:40,41: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎
∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥
)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢
([(𝑎 ∩
𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏 (𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) |
| |
| Theorem | onfrALTlem4VD 44858* |
Virtual deduction proof of onfrALTlem4 44516.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem4 44516 is onfrALTlem4VD 44858 without virtual deductions and was
automatically derived from onfrALTlem4VD 44858.
| 1:: | ⊢ 𝑦 ∈ V
| | 2:1: | ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋
𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅)
| | 3:1: | ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌
𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥)
| | 4:1: | ⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎
| | 5:1: | ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦
| | 6:4,5: | ⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = (
𝑎 ∩ 𝑦)
| | 7:3,6: | ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦)
| | 8:1: | ⊢ ⦋𝑦 / 𝑥⦌∅ = ∅
| | 9:7,8: | ⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌
∅ ↔ (𝑎 ∩ 𝑦) = ∅)
| | 10:2,9: | ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎
∩ 𝑦) = ∅)
| | 11:1: | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎)
| | 12:11,10: | ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](
𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| | 13:1: | ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) =
∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅))
| | qed:13,12: | ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) =
∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
| |
| Theorem | onfrALTlem3VD 44859* |
Virtual deduction proof of onfrALTlem3 44517.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem3 44517 is onfrALTlem3VD 44859 without virtual deductions and was
automatically derived from onfrALTlem3VD 44859.
| 1:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑎
⊆ On ∧ 𝑎 ≠ ∅) )
| | 2:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) )
| | 3:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ 𝑥 ∈ 𝑎 )
| | 4:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ 𝑎 ⊆
On )
| | 5:3,4: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ 𝑥 ∈ On )
| | 6:5: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ Ord 𝑥 )
| | 7:6: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ E We 𝑥 )
| | 8:: | ⊢ (𝑎 ∩ 𝑥) ⊆ 𝑥
| | 9:7,8: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ E We (𝑎 ∩ 𝑥) )
| | 10:9: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ E Fr (𝑎 ∩ 𝑥) )
| | 11:10: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∀𝑏((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠
∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) )
| | 12:: | ⊢ 𝑥 ∈ V
| | 13:12,8: | ⊢ (𝑎 ∩ 𝑥) ∈ V
| | 14:13,11: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ [(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) )
| | 15:: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩
𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)(
(𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
| | 16:14,15: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (
𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) =
∅) )
| | 17:: | ⊢ (𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥)
| | 18:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ¬ (𝑎 ∩ 𝑥) = ∅ )
| | 19:18: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (𝑎 ∩ 𝑥) ≠ ∅ )
| | 20:17,19: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩
𝑥) ≠ ∅) )
| | qed:16,20: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅ )
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ ) |
| |
| Theorem | simplbi2comtVD 44860 |
Virtual deduction proof of simplbi2comt 501.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
simplbi2comt 501 is simplbi2comtVD 44860 without virtual deductions and was
automatically derived from simplbi2comtVD 44860.
| 1:: | ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜑 ↔ (
𝜓 ∧ 𝜒)) )
| | 2:1: | ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ ((𝜓 ∧ 𝜒
) → 𝜑) )
| | 3:2: | ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜓 → (𝜒
→ 𝜑)) )
| | 4:3: | ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜒 → (𝜓
→ 𝜑)) )
| | qed:4: | ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓
→ 𝜑)))
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑))) |
| |
| Theorem | onfrALTlem2VD 44861* |
Virtual deduction proof of onfrALTlem2 44519.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem2 44519 is onfrALTlem2VD 44861 without virtual deductions and was
automatically derived from onfrALTlem2VD 44861.
| 1:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩
𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) )
| | 2:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ (𝑎 ∩ 𝑦) )
| | 3:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ 𝑎 )
| | 4:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑎
⊆ On ∧ 𝑎 ≠ ∅) )
| | 5:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) )
| | 6:5: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ 𝑥 ∈ 𝑎 )
| | 7:4: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ 𝑎 ⊆
On )
| | 8:6,7: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ 𝑥 ∈ On )
| | 9:8: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ Ord 𝑥 )
| | 10:9: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ Tr 𝑥 )
| | 11:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑦 ∈ (𝑎 ∩ 𝑥) )
| | 12:11: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑦 ∈ 𝑥 )
| | 13:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ 𝑦 )
| | 14:10,12,13: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ 𝑥 )
| | 15:3,14: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ (𝑎 ∩ 𝑥) )
| | 16:13,15: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦) )
| | 17:16: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)) )
| | 18:17: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ ∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)) )
| | 19:18: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦) )
| | 20:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) )
| | 21:20: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ )
| | 22:19,21: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑎 ∩ 𝑦) = ∅ )
| | 23:20: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ 𝑦 ∈ (𝑎 ∩ 𝑥) )
| | 24:23: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ 𝑦 ∈ 𝑎 )
| | 25:22,24: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅) )
| | 26:25: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥)
∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) )
| | 27:26: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∀𝑦((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥
) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) )
| | 28:27: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥
) ∩ 𝑦) = ∅) → ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) )
| | 29:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅ )
| | 30:29: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥)
∩ 𝑦) = ∅) )
| | 31:28,30: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅) )
| | qed:31: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ) |
| |
| Theorem | onfrALTlem1VD 44862* |
Virtual deduction proof of onfrALTlem1 44521.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem1 44521 is onfrALTlem1VD 44862 without virtual deductions and was
automatically derived from onfrALTlem1VD 44862.
| 1:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) )
| | 2:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) )
| | 3:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)
)
| | 4:: | ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅
) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| | 5:4: | ⊢ ∀𝑦([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥)
= ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| | 6:5: | ⊢ (∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥)
= ∅) ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| | 7:3,6: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅) )
| | 8:: | ⊢ (∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦(
𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| | qed:7,8: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ) |
| |
| Theorem | onfrALTVD 44863 |
Virtual deduction proof of onfrALT 44522.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALT 44522 is onfrALTVD 44863 without virtual deductions and was
automatically derived from onfrALTVD 44863.
| 1:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎
∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
| | 2:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎
∧ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
| | 3:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , 𝑥 ∈ 𝑎 ▶
(¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| | 4:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , 𝑥 ∈ 𝑎 ▶
((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| | 5:: | ⊢ ((𝑎 ∩ 𝑥) = ∅ ∨ ¬ (𝑎 ∩ 𝑥) =
∅)
| | 6:5,4,3: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , 𝑥 ∈ 𝑎 ▶
∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
| | 7:6: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑥 ∈ 𝑎
→ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| | 8:7: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ ∀𝑥(𝑥
∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| | 9:8: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (∃𝑥𝑥
∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| | 10:: | ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥𝑥 ∈ 𝑎)
| | 11:9,10: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑎 ≠
∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| | 12:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑎 ⊆
On ∧ 𝑎 ≠ ∅) )
| | 13:12: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ 𝑎 ≠
∅ )
| | 14:13,11: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ ∃𝑦 ∈
𝑎(𝑎 ∩ 𝑦) = ∅ )
| | 15:14: | ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎
(𝑎 ∩ 𝑦) = ∅)
| | 16:15: | ⊢ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦
∈ 𝑎(𝑎 ∩ 𝑦) = ∅)
| | qed:16: | ⊢ E Fr On
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ E Fr
On |
| |
| Theorem | csbeq2gVD 44864 |
Virtual deduction proof of csbeq2 3879.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbeq2 3879 is csbeq2gVD 44864 without virtual deductions and was
automatically derived from csbeq2gVD 44864.
| 1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∀𝑥𝐵 = 𝐶 → [𝐴 / 𝑥]
𝐵 = 𝐶) )
| | 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴
/ 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) )
| | 4:2,3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥
⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) )
| | qed:4: | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌
𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
| |
| Theorem | csbsngVD 44865 |
Virtual deduction proof of csbsng 4684.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbsng 4684 is csbsngVD 44865 without virtual deductions and was automatically
derived from csbsngVD 44865.
| 1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 = 𝐵
↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵) )
| | 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌𝑦 = 𝑦 )
| | 4:3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴
/ 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) )
| | 5:2,4: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 = 𝐵
↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) )
| | 6:5: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑦([𝐴 / 𝑥]𝑦
= 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) )
| | 7:6: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑦 ∣ [𝐴 / 𝑥]𝑦 =
𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} )
| | 8:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑦 ∣ [𝐴 / 𝑥]𝑦 =
𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} )
| | 9:7,8: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦
= 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} )
| | 10:: | ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
| | 11:10: | ⊢ ∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
| | 12:1,11: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝐵} = ⦋
𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} )
| | 13:9,12: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝐵} = {
𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} )
| | 14:: | ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴
/ 𝑥⦌𝐵}
| | 15:13,14: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝐵} = {
⦋𝐴 / 𝑥⦌𝐵} )
| | qed:15: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋
𝐴 / 𝑥⦌𝐵})
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}) |
| |
| Theorem | csbxpgVD 44866 |
Virtual deduction proof of csbxp 5754.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbxp 5754 is csbxpgVD 44866 without virtual deductions and was
automatically derived from csbxpgVD 44866.
| 1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔
⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌𝑤 = 𝑤 )
| | 4:3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 /
𝑥⦌𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 5:2,4: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤
∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 6:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔
⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) )
| | 7:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌𝑦 = 𝑦 )
| | 8:7: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 /
𝑥⦌𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) )
| | 9:6,8: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐶) )
| | 10:5,9: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧
[𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)) )
| | 11:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧
𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) )
| | 12:10,11: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧
𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)) )
| | 13:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑧 = ⟨𝑤 ,
𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩) )
| | 14:12,13: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (([𝐴 / 𝑥]𝑧 = ⟨𝑤
, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩
∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| | 15:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑧 = ⟨𝑤
, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩
∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) )
| | 16:14,15: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑧 = ⟨𝑤
, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| | 17:16: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑦([𝐴 / 𝑥](𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| | 18:17: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑦[𝐴 / 𝑥](𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| | 19:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑦(𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) )
| | 20:18,19: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑦(𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| | 21:20: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑤([𝐴 / 𝑥]∃𝑦(
𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| | 22:21: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑤[𝐴 / 𝑥]∃𝑦(
𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| | 23:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑤∃𝑦(
𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦
(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) )
| | 24:22,23: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑤∃𝑦(
𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| | 25:24: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑧([𝐴 / 𝑥]∃𝑤∃
𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| | 26:25: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃
𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(
𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
)
| | 27:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃
𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥]
∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} )
| | 28:26,27: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃
𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(
𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
)
| | 29:: | ⊢ {⟨𝑤 , 𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
= {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
| | 30:: | ⊢ (𝐵 × 𝐶) = {⟨𝑤 , 𝑦⟩ ∣ (𝑤 ∈ 𝐵
∧ 𝑦 ∈ 𝐶)}
| | 31:29,30: | ⊢ (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤
, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
| | 32:31: | ⊢ ∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 =
⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
| | 33:1,32: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) =
⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧
𝑦 ∈ 𝐶))} )
| | 34:28,33: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) =
{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} )
| | 35:: | ⊢ {⟨𝑤 , 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
| | 36:: | ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {
⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)}
| | 37:35,36: | ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {𝑧
∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
| | 38:34,37: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) =
(⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) )
| | qed:38: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (
⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶)) |
| |
| Theorem | csbresgVD 44867 |
Virtual deduction proof of csbres 5969.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbres 5969 is csbresgVD 44867 without virtual deductions and was
automatically derived from csbresgVD 44867.
| 1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌V = V )
| | 3:2: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 /
𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V) )
| | 4:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐶 × V) =
(⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) )
| | 5:3,4: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐶 × V) =
(⦋𝐴 / 𝑥⦌𝐶 × V) )
| | 6:5: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 /
𝑥⦌(𝐶 × V)) =
(⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) )
| | 7:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 ×
V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) )
| | 8:6,7: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 ×
V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) )
| | 9:: | ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
| | 10:9: | ⊢ ∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
| | 11:1,10: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) =
⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) )
| | 12:8,11: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶)
= (
⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) )
| | 13:: | ⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (
⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))
| | 14:12,13: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) =
(
⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) )
| | qed:14: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (
⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)) |
| |
| Theorem | csbrngVD 44868 |
Virtual deduction proof of csbrn 6192.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbrn 6192 is csbrngVD 44868 without virtual deductions and was
automatically derived from csbrngVD 44868.
| 1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]⟨𝑤 , 𝑦⟩
∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌⟨𝑤 , 𝑦⟩ =
⟨𝑤, 𝑦⟩ )
| | 4:3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌⟨𝑤 , 𝑦⟩
∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 5:2,4: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]⟨𝑤 , 𝑦⟩
∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 6:5: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑤([𝐴 / 𝑥]⟨𝑤 ,
𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 7:6: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑤[𝐴 / 𝑥]⟨𝑤 ,
𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 8:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑤[𝐴 / 𝑥]⟨𝑤 ,
𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵) )
| | 9:7,8: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑤⟨𝑤
, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 10:9: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑦([𝐴 / 𝑥]∃𝑤
⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 11:10: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨
𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} )
| | 12:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤
⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} )
| | 13:11,12: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤
⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} )
| | 14:: | ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤 , 𝑦⟩ ∈ 𝐵}
| | 15:14: | ⊢ ∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤 , 𝑦⟩
∈ 𝐵}
| | 16:1,15: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 /
𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} )
| | 17:13,16: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣
∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} )
| | 18:: | ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤
, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
| | 19:17,18: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋
𝐴 / 𝑥⦌𝐵 )
| | qed:19: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴
/ 𝑥⦌𝐵)
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵) |
| |
| Theorem | csbima12gALTVD 44869 |
Virtual deduction proof of csbima12 6066.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbima12 6066 is csbima12gALTVD 44869 without virtual deductions and was
automatically derived from csbima12gALTVD 44869.
| 1:: | ⊢ ( 𝐴 ∈ 𝐶 ▶ 𝐴 ∈ 𝐶 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) =
(
⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) )
| | 3:2: | ⊢ ( 𝐴 ∈ 𝐶 ▶
ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵)
= ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) )
| | 4:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶
⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)
= ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) )
| | 5:3,4: | ⊢ ( 𝐴 ∈ 𝐶 ▶
⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)
= ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) )
| | 6:: | ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
| | 7:6: | ⊢ ∀𝑥(𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
| | 8:1,7: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋
𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) )
| | 9:5,8: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) =
ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) )
| | 10:: | ⊢ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = ran
(⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)
| | 11:9,10: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (
⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) )
| | qed:11: | ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋
𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)) |
| |
| Theorem | csbunigVD 44870 |
Virtual deduction proof of csbuni 4912.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbuni 4912 is csbunigVD 44870 without virtual deductions and was
automatically derived from csbunigVD 44870.
| 1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧
∈ 𝑦) )
| | 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐵) )
| | 4:2,3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧
[𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| | 5:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧
𝑦 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵)) )
| | 6:4,5: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧
𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| | 7:6: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑦([𝐴 / 𝑥](𝑧 ∈
𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| | 8:7: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑦[𝐴 / 𝑥](𝑧 ∈
𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| | 9:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑦(𝑧 ∈
𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) )
| | 10:8,9: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑦(𝑧 ∈
𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| | 11:10: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑧([𝐴 / 𝑥]∃𝑦(
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| | 12:11: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)} )
| | 13:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
)
| | 14:12,13: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)} )
| | 15:: | ⊢ ∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
| | 16:15: | ⊢ ∀𝑥∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈
𝐵)}
| | 17:1,16: | ⊢ ( 𝐴 ∈ 𝑉 ▶ [𝐴 / 𝑥]∪ 𝐵 = {𝑧 ∣
∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} )
| | 18:1,17: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ⦋𝐴 /
𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} )
| | 19:14,18: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌∪ 𝐵 = {𝑧 ∣
∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)} )
| | 20:: | ⊢ ∪ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦
∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}
| | 21:19,20: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴
/ 𝑥⦌𝐵 )
| | qed:21: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 /
𝑥⦌𝐵)
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪
𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵) |
| |
| Theorem | csbfv12gALTVD 44871 |
Virtual deduction proof of csbfv12 6923.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbfv12 6923 is csbfv12gALTVD 44871 without virtual deductions and was
automatically derived from csbfv12gALTVD 44871.
| 1:: | ⊢ ( 𝐴 ∈ 𝐶 ▶ 𝐴 ∈ 𝐶 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌{𝑦} = {
𝑦} )
| | 3:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵
}) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) )
| | 4:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌{𝐵} = {
⦋𝐴 / 𝑥⦌𝐵} )
| | 5:4: | ⊢ ( 𝐴 ∈ 𝐶 ▶ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴
/ 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) )
| | 6:3,5: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵
}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) )
| | 7:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ([𝐴 / 𝑥](𝐹 “ {
𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦}) )
| | 8:6,2: | ⊢ ( 𝐴 ∈ 𝐶 ▶ (⦋𝐴 / 𝑥⦌(𝐹 “ {
𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})
= {𝑦}) )
| | 9:7,8: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ([𝐴 / 𝑥](𝐹 “ {
𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})
)
| | 10:9: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ∀𝑦([𝐴 / 𝑥](𝐹
“ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦}) )
| | 11:10: | ⊢ ( 𝐴 ∈ 𝐶 ▶ {𝑦 ∣ [𝐴 / 𝑥](𝐹
“ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦}} )
| | 12:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹
“ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} )
| | 13:11,12: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹
“ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦
}} )
| | 14:13: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (
𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “
{⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦}} )
| | 15:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (
𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) =
{𝑦}} )
| | 16:14,15: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (
𝐹 “ {𝐵}) = {𝑦}} =
∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦}} )
| | 17:: | ⊢ (𝐹‘𝐵) =
∪ {𝑦 ∣ (𝐹 “ {𝐵}) =
{𝑦}}
| | 18:17: | ⊢ ∀𝑥(𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵
}) = {𝑦}}
| | 19:1,18: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵)
= ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} )
| | 20:16,19: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵)
= ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} )
| | 21:: | ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) =
∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}
| | 22:20,21: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵)
= (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) )
| | qed:22: | ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) =
(⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)) |
| |
| Theorem | con5VD 44872 |
Virtual deduction proof of con5 44495.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
con5 44495 is con5VD 44872 without virtual deductions and was automatically
derived from con5VD 44872.
| 1:: | ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (𝜑 ↔ ¬ 𝜓) )
| | 2:1: | ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜓 → 𝜑) )
| | 3:2: | ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → ¬ ¬ 𝜓
) )
| | 4:: | ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
| | 5:3,4: | ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → 𝜓) )
| | qed:5: | ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
| |
| Theorem | relopabVD 44873 |
Virtual deduction proof of relopab 5803.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
relopab 5803 is relopabVD 44873 without virtual deductions and was
automatically derived from relopabVD 44873.
| 1:: | ⊢ ( 𝑦 = 𝑣 ▶ 𝑦 = 𝑣 )
| | 2:1: | ⊢ ( 𝑦 = 𝑣 ▶ ⟨𝑥 , 𝑦⟩ = ⟨𝑥 , 𝑣
⟩ )
| | 3:: | ⊢ ( 𝑦 = 𝑣 , 𝑥 = 𝑢 ▶ 𝑥 = 𝑢 )
| | 4:3: | ⊢ ( 𝑦 = 𝑣 , 𝑥 = 𝑢 ▶ ⟨𝑥 , 𝑣⟩ = ⟨
𝑢, 𝑣⟩ )
| | 5:2,4: | ⊢ ( 𝑦 = 𝑣 , 𝑥 = 𝑢 ▶ ⟨𝑥 , 𝑦⟩ = ⟨
𝑢, 𝑣⟩ )
| | 6:5: | ⊢ ( 𝑦 = 𝑣 , 𝑥 = 𝑢 ▶ (𝑧 = ⟨𝑥 , 𝑦
⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) )
| | 7:6: | ⊢ ( 𝑦 = 𝑣 ▶ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥 ,
𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)) )
| | 8:7: | ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥 , 𝑦
⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
| | 9:8: | ⊢ (∃𝑣𝑦 = 𝑣 → ∃𝑣(𝑥 = 𝑢 → (𝑧
= ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
| | 90:: | ⊢ (𝑣 = 𝑦 ↔ 𝑦 = 𝑣)
| | 91:90: | ⊢ (∃𝑣𝑣 = 𝑦 ↔ ∃𝑣𝑦 = 𝑣)
| | 92:: | ⊢ ∃𝑣𝑣 = 𝑦
| | 10:91,92: | ⊢ ∃𝑣𝑦 = 𝑣
| | 11:9,10: | ⊢ ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥 , 𝑦⟩ →
𝑧 = ⟨𝑢, 𝑣⟩))
| | 12:11: | ⊢ (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥 , 𝑦⟩ →
𝑧 = ⟨𝑢, 𝑣⟩))
| | 13:: | ⊢ (∃𝑣(𝑧 = ⟨𝑥 , 𝑦⟩ → 𝑧 = ⟨𝑢
, 𝑣⟩) → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩))
| | 14:12,13: | ⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥 , 𝑦⟩ → ∃𝑣
𝑧 = ⟨𝑢, 𝑣⟩))
| | 15:14: | ⊢ (∃𝑢𝑥 = 𝑢 → ∃𝑢(𝑧 = ⟨𝑥 , 𝑦
⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩))
| | 150:: | ⊢ (𝑢 = 𝑥 ↔ 𝑥 = 𝑢)
| | 151:150: | ⊢ (∃𝑢𝑢 = 𝑥 ↔ ∃𝑢𝑥 = 𝑢)
| | 152:: | ⊢ ∃𝑢𝑢 = 𝑥
| | 16:151,152: | ⊢ ∃𝑢𝑥 = 𝑢
| | 17:15,16: | ⊢ ∃𝑢(𝑧 = ⟨𝑥 , 𝑦⟩ → ∃𝑣𝑧 = ⟨
𝑢, 𝑣⟩)
| | 18:17: | ⊢ (𝑧 = ⟨𝑥 , 𝑦⟩ → ∃𝑢∃𝑣𝑧 = ⟨
𝑢, 𝑣⟩)
| | 19:18: | ⊢ (∃𝑦𝑧 = ⟨𝑥 , 𝑦⟩ → ∃𝑦∃𝑢
∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
| | 20:: | ⊢ (∃𝑦∃𝑢∃𝑣𝑧 = ⟨𝑢 , 𝑣⟩ →
∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
| | 21:19,20: | ⊢ (∃𝑦𝑧 = ⟨𝑥 , 𝑦⟩ → ∃𝑢∃𝑣𝑧
= ⟨𝑢, 𝑣⟩)
| | 22:21: | ⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥 , 𝑦⟩ → ∃𝑥
∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
| | 23:: | ⊢ (∃𝑥∃𝑢∃𝑣𝑧 = ⟨𝑢 , 𝑣⟩ →
∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
| | 24:22,23: | ⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥 , 𝑦⟩ → ∃𝑢
∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
| | 25:24: | ⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = ⟨𝑥 , 𝑦⟩} ⊆
{𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩}
| | 26:: | ⊢ 𝑥 ∈ V
| | 27:: | ⊢ 𝑦 ∈ V
| | 28:26,27: | ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
| | 29:28: | ⊢ (𝑧 = ⟨𝑥 , 𝑦⟩ ↔ (𝑧 = ⟨𝑥 , 𝑦
⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
| | 30:29: | ⊢ (∃𝑦𝑧 = ⟨𝑥 , 𝑦⟩ ↔ ∃𝑦(𝑧 =
⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
| | 31:30: | ⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥 , 𝑦⟩ ↔ ∃𝑥
∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
| | 32:31: | ⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = ⟨𝑥 , 𝑦⟩} = {
𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
| | 320:25,32: | ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥 , 𝑦⟩ ∧
(𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩}
| | 33:: | ⊢ 𝑢 ∈ V
| | 34:: | ⊢ 𝑣 ∈ V
| | 35:33,34: | ⊢ (𝑢 ∈ V ∧ 𝑣 ∈ V)
| | 36:35: | ⊢ (𝑧 = ⟨𝑢 , 𝑣⟩ ↔ (𝑧 = ⟨𝑢 , 𝑣
⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
| | 37:36: | ⊢ (∃𝑣𝑧 = ⟨𝑢 , 𝑣⟩ ↔ ∃𝑣(𝑧 =
⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
| | 38:37: | ⊢ (∃𝑢∃𝑣𝑧 = ⟨𝑢 , 𝑣⟩ ↔ ∃𝑢
∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
| | 39:38: | ⊢ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢 , 𝑣⟩} = {
𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
| | 40:320,39: | ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥 , 𝑦⟩ ∧
(𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧
(𝑢 ∈ V ∧ 𝑣 ∈ V))}
| | 41:: | ⊢ {⟨𝑥 , 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V
)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))
}
| | 42:: | ⊢ {⟨𝑢 , 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V
)} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))
}
| | 43:40,41,42: | ⊢ {⟨𝑥 , 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V
)} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
| | 44:: | ⊢ {⟨𝑢 , 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V
)} = (V × V)
| | 45:43,44: | ⊢ {⟨𝑥 , 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V
)} ⊆ (V × V)
| | 46:28: | ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
| | 47:46: | ⊢ {⟨𝑥 , 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥 , 𝑦⟩
∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
| | 48:45,47: | ⊢ {⟨𝑥 , 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
| | qed:48: | ⊢ Rel {⟨𝑥 , 𝑦⟩ ∣ 𝜑}
|
(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ Rel
{⟨𝑥, 𝑦⟩ ∣ 𝜑} |
| |
| Theorem | 19.41rgVD 44874 |
Virtual deduction proof of 19.41rg 44523.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. 19.41rg 44523
is 19.41rgVD 44874 without virtual deductions and was automatically derived
from 19.41rgVD 44874. (Contributed by Alan Sare, 8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
| | 2:1: | ⊢ ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (
𝜑 ∧ 𝜓))))
| | 3:2: | ⊢ ∀𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑
→ (𝜑 ∧ 𝜓))))
| | 4:3: | ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 →
∀𝑥(𝜑 → (𝜑 ∧ 𝜓))))
| | 5:: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ ∀𝑥(𝜓
→ ∀𝑥𝜓) )
| | 6:4,5: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (∀𝑥𝜓
→ ∀𝑥(𝜑 → (𝜑 ∧ 𝜓))) )
| | 7:: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) , ∀𝑥𝜓 ▶
∀𝑥𝜓 )
| | 8:6,7: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) , ∀𝑥𝜓 ▶
∀𝑥(𝜑 → (𝜑 ∧ 𝜓)) )
| | 9:8: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) , ∀𝑥𝜓 ▶
(∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)) )
| | 10:9: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (∀𝑥𝜓
→ (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) )
| | 11:5: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (𝜓 → ∀
𝑥𝜓) )
| | 12:10,11: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (𝜓 → (
∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) )
| | 13:12: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (∃𝑥𝜑
→ (𝜓 → ∃𝑥(𝜑 ∧ 𝜓))) )
| | 14:13: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ ((∃𝑥
𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) )
| | qed:14: | ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑
∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
|
|
| ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) |
| |
| Theorem | 2pm13.193VD 44875 |
Virtual deduction proof of 2pm13.193 44525.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
2pm13.193 44525 is 2pm13.193VD 44875 without virtual deductions and was
automatically derived from 2pm13.193VD 44875. (Contributed by Alan Sare,
8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) )
| | 2:1: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) )
| | 3:2: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ 𝑥 = 𝑢 )
| | 4:1: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑 )
| | 5:3,4: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) )
| | 6:5: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) )
| | 7:6: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ [𝑣 / 𝑦]𝜑 )
| | 8:2: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ 𝑦 = 𝑣 )
| | 9:7,8: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣) )
| | 10:9: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ (𝜑 ∧ 𝑦 = 𝑣) )
| | 11:10: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ 𝜑 )
| | 12:2,11: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) )
| | 13:12: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣
/ 𝑦]𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 14:: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ((
𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) )
| | 15:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ (𝑥
= 𝑢 ∧ 𝑦 = 𝑣) )
| | 16:15: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ 𝑦 =
𝑣 )
| | 17:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ 𝜑
)
| | 18:16,17: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ (
𝜑 ∧ 𝑦 = 𝑣) )
| | 19:18: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ([
𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣) )
| | 20:15: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ 𝑥 =
𝑢 )
| | 21:19: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ [𝑣
/ 𝑦]𝜑 )
| | 22:20,21: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ([
𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) )
| | 23:22: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ([
𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) )
| | 24:23: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ [𝑢
/ 𝑥][𝑣 / 𝑦]𝜑 )
| | 25:15,24: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ((
𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) )
| | 26:25: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → ((𝑥
= 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| | qed:13,26: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣
/ 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
|
|
| ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) |
| |
| Theorem | hbimpgVD 44876 |
Virtual deduction proof of hbimpg 44527.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbimpg 44527
is hbimpgVD 44876 without virtual deductions and was automatically derived
from hbimpgVD 44876. (Contributed by Alan Sare, 8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 →
∀𝑥𝜓)) )
| | 2:1: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ∀𝑥(𝜑 → ∀𝑥𝜑) )
| | 3:: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)), ¬ 𝜑 ▶ ¬ 𝜑 )
| | 4:2: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ∀𝑥(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| | 5:4: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (¬ 𝜑 → ∀𝑥¬ 𝜑) )
| | 6:3,5: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)), ¬ 𝜑 ▶ ∀𝑥¬ 𝜑 )
| | 7:: | ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
| | 8:7: | ⊢ (∀𝑥¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
| | 9:6,8: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)), ¬ 𝜑 ▶ ∀𝑥(𝜑 → 𝜓) )
| | 10:9: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (¬ 𝜑 → ∀𝑥(𝜑 → 𝜓)) )
| | 11:: | ⊢ (𝜓 → (𝜑 → 𝜓))
| | 12:11: | ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓))
| | 13:1: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ∀𝑥(𝜓 → ∀𝑥𝜓) )
| | 14:13: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (𝜓 → ∀𝑥𝜓) )
| | 15:14,12: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (𝜓 → ∀𝑥(𝜑 → 𝜓)) )
| | 16:10,15: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ((¬ 𝜑 ∨ 𝜓) → ∀𝑥(𝜑 → 𝜓)) )
| | 17:: | ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
| | 18:16,17: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) )
| | 19:: | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥(
𝜑 → ∀𝑥𝜑))
| | 20:: | ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ∀𝑥∀𝑥(
𝜓 → ∀𝑥𝜓))
| | 21:19,20: | ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) → ∀𝑥(∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 →
∀𝑥𝜓)))
| | 22:21,18: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) )
| | qed:22: | ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
|
|
| ⊢
((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))) |
| |
| Theorem | hbalgVD 44877 |
Virtual deduction proof of hbalg 44528.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbalg 44528
is hbalgVD 44877 without virtual deductions and was automatically derived
from hbalgVD 44877. (Contributed by Alan Sare, 8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦(𝜑
→ ∀𝑥𝜑) )
| | 2:1: | ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∀𝑦𝜑
→ ∀𝑦∀𝑥𝜑) )
| | 3:: | ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
| | 4:2,3: | ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∀𝑦𝜑
→ ∀𝑥∀𝑦𝜑) )
| | 5:: | ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦(
𝜑 → ∀𝑥𝜑))
| | 6:5,4: | ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦(∀
𝑦𝜑 → ∀𝑥∀𝑦𝜑) )
| | qed:6: | ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦
𝜑 → ∀𝑥∀𝑦𝜑))
|
|
| ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) |
| |
| Theorem | hbexgVD 44878 |
Virtual deduction proof of hbexg 44529.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbexg 44529
is hbexgVD 44878 without virtual deductions and was automatically derived
from hbexgVD 44878. (Contributed by Alan Sare, 8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
∀𝑦(𝜑 → ∀𝑥𝜑) )
| | 2:1: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
∀𝑥(𝜑 → ∀𝑥𝜑) )
| | 3:2: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
(𝜑 → ∀𝑥𝜑) )
| | 4:3: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| | 5:: | ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦
∀𝑥(𝜑 → ∀𝑥𝜑))
| | 6:: | ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦
∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
| | 7:5: | ⊢ (∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔
∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
| | 8:5,6,7: | ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦
∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
| | 9:8,4: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
∀𝑥(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| | 10:9: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
∀𝑦(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| | 11:10: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| | 12:11: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
(∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑) )
| | 13:12: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∀
𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑) )
| | 14:: | ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥
∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
| | 15:13,14: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
(∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑) )
| | 16:15: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
(¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑) )
| | 17:16: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (¬
∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑) )
| | 18:: | ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦¬ 𝜑)
| | 19:17,18: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∃
𝑦𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑) )
| | 20:18: | ⊢ (∀𝑥∃𝑦𝜑 ↔ ∀𝑥¬ ∀𝑦¬ 𝜑)
| | 21:19,20: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∃
𝑦𝜑 → ∀𝑥∃𝑦𝜑) )
| | 22:8,21: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) )
| | 23:14,22: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) )
| | qed:23: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) )
|
|
| ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) |
| |
| Theorem | ax6e2eqVD 44879* |
The following User's Proof is a Virtual Deduction proof (see wvd1 44542)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 44530 is ax6e2eqVD 44879 without virtual
deductions and was automatically derived from ax6e2eqVD 44879.
(Contributed by Alan Sare, 25-Mar-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥𝑥 = 𝑦 )
| | 2:: | ⊢ ( ∀𝑥𝑥 = 𝑦 , 𝑥 = 𝑢 ▶ 𝑥 = 𝑢 )
| | 3:1: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ 𝑥 = 𝑦 )
| | 4:2,3: | ⊢ ( ∀𝑥𝑥 = 𝑦 , 𝑥 = 𝑢 ▶ 𝑦 = 𝑢 )
| | 5:2,4: | ⊢ ( ∀𝑥𝑥 = 𝑦 , 𝑥 = 𝑢 ▶ (𝑥 = 𝑢 ∧ 𝑦
= 𝑢) )
| | 6:5: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧
𝑦 = 𝑢)) )
| | 7:6: | ⊢ (∀𝑥𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦
= 𝑢)))
| | 8:7: | ⊢ (∀𝑥∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (
𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
| | 9:: | ⊢ (∀𝑥𝑥 = 𝑦 ↔ ∀𝑥∀𝑥𝑥 = 𝑦)
| | 10:8,9: | ⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢
∧ 𝑦 = 𝑢)))
| | 11:1,10: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥(𝑥 = 𝑢 → (𝑥 =
𝑢 ∧ 𝑦 = 𝑢)) )
| | 12:11: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ (∃𝑥𝑥 = 𝑢 → ∃𝑥
(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)) )
| | 13:: | ⊢ ∃𝑥𝑥 = 𝑢
| | 14:13,12: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢
) )
| | 140:14: | ⊢ (∀𝑥𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)
)
| | 141:140: | ⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦
= 𝑢))
| | 15:1,141: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑢) )
| | 16:1,15: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑦∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑢) )
| | 17:16: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∃𝑦∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑢) )
| | 18:17: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧
𝑦 = 𝑢) )
| | 19:: | ⊢ ( 𝑢 = 𝑣 ▶ 𝑢 = 𝑣 )
| | 20:: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ (𝑥 =
𝑢 ∧ 𝑦 = 𝑢) )
| | 21:20: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ 𝑦 = 𝑢
)
| | 22:19,21: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ 𝑦 = 𝑣
)
| | 23:20: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ 𝑥 = 𝑢
)
| | 24:22,23: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ (𝑥 =
𝑢 ∧ 𝑦 = 𝑣) )
| | 25:24: | ⊢ ( 𝑢 = 𝑣 ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → (
𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| | 26:25: | ⊢ ( 𝑢 = 𝑣 ▶ ∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑢)
→ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| | 27:26: | ⊢ ( 𝑢 = 𝑣 ▶ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)
→ ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| | 28:27: | ⊢ ( 𝑢 = 𝑣 ▶ ∀𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 =
𝑢) → ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| | 29:28: | ⊢ ( 𝑢 = 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 =
𝑢) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| | 30:29: | ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢
) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| | 31:18,30: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ (𝑢 = 𝑣 → ∃𝑥∃𝑦
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| | qed:31: | ⊢ (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(
𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
|
|
| ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) |
| |
| Theorem | ax6e2ndVD 44880* |
The following User's Proof is a Virtual Deduction proof (see wvd1 44542)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2nd 44531 is ax6e2ndVD 44880 without virtual
deductions and was automatically derived from ax6e2ndVD 44880.
(Contributed by Alan Sare, 25-Mar-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ∃𝑦𝑦 = 𝑣
| | 2:: | ⊢ 𝑢 ∈ V
| | 3:1,2: | ⊢ (𝑢 ∈ V ∧ ∃𝑦𝑦 = 𝑣)
| | 4:3: | ⊢ ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣)
| | 5:: | ⊢ (𝑢 ∈ V ↔ ∃𝑥𝑥 = 𝑢)
| | 6:5: | ⊢ ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥𝑥 =
𝑢 ∧ 𝑦 = 𝑣))
| | 7:6: | ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦
(∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
| | 8:4,7: | ⊢ ∃𝑦(∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣)
| | 9:: | ⊢ (𝑧 = 𝑣 → ∀𝑥𝑧 = 𝑣)
| | 10:: | ⊢ (𝑦 = 𝑣 → ∀𝑧𝑦 = 𝑣)
| | 11:: | ⊢ ( 𝑧 = 𝑦 ▶ 𝑧 = 𝑦 )
| | 12:11: | ⊢ ( 𝑧 = 𝑦 ▶ (𝑧 = 𝑣 ↔ 𝑦 = 𝑣) )
| | 120:11: | ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
| | 13:9,10,120: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥𝑦
= 𝑣))
| | 14:: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ¬ ∀𝑥𝑥 = 𝑦 )
| | 15:14,13: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ (𝑦 = 𝑣 → ∀𝑥
𝑦 = 𝑣) )
| | 16:15: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥𝑦
= 𝑣))
| | 17:16: | ⊢ (∀𝑥¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣
→ ∀𝑥𝑦 = 𝑣))
| | 18:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦
)
| | 19:17,18: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀
𝑥𝑦 = 𝑣))
| | 20:14,19: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥(𝑦 = 𝑣 →
∀𝑥𝑦 = 𝑣) )
| | 21:20: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ((∃𝑥𝑥 = 𝑢
∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| | 22:21: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ((∃𝑥𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| | 23:22: | ⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥
𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| | 24:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦
)
| | 25:23,24: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥𝑥 =
𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| | 26:14,25: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ∀𝑦((∃𝑥𝑥
= 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| | 27:26: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ (∃𝑦(∃𝑥𝑥
= 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| | 28:8,27: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ∃𝑦∃𝑥(𝑥 =
𝑢 ∧ 𝑦 = 𝑣) )
| | 29:28: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ∃𝑥∃𝑦(𝑥 =
𝑢 ∧ 𝑦 = 𝑣) )
| | qed:29: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢
∧ 𝑦 = 𝑣))
|
|
| ⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
| |
| Theorem | ax6e2ndeqVD 44881* |
The following User's Proof is a Virtual Deduction proof (see wvd1 44542)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 44530 is ax6e2ndeqVD 44881 without virtual
deductions and was automatically derived from ax6e2ndeqVD 44881.
(Contributed by Alan Sare, 25-Mar-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( 𝑢 ≠ 𝑣 ▶ 𝑢 ≠ 𝑣 )
| | 2:: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ (
𝑥 = 𝑢 ∧ 𝑦 = 𝑣) )
| | 3:2: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ 𝑥
= 𝑢 )
| | 4:1,3: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ 𝑥
≠ 𝑣 )
| | 5:2: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ 𝑦
= 𝑣 )
| | 6:4,5: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ 𝑥
≠ 𝑦 )
| | 7:: | ⊢ (∀𝑥𝑥 = 𝑦 → 𝑥 = 𝑦)
| | 8:7: | ⊢ (¬ 𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
| | 9:: | ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
| | 10:8,9: | ⊢ (𝑥 ≠ 𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
| | 11:6,10: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶
¬ ∀𝑥𝑥 = 𝑦 )
| | 12:11: | ⊢ ( 𝑢 ≠ 𝑣 ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣)
→ ¬ ∀𝑥𝑥 = 𝑦) )
| | 13:12: | ⊢ ( 𝑢 ≠ 𝑣 ▶ ∀𝑥((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) → ¬ ∀𝑥𝑥 = 𝑦) )
| | 14:13: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 =
𝑣) → ∃𝑥¬ ∀𝑥𝑥 = 𝑦) )
| | 15:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦
)
| | 19:15: | ⊢ (∃𝑥¬ ∀𝑥𝑥 = 𝑦 ↔ ¬ ∀𝑥𝑥 =
𝑦)
| | 20:14,19: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 =
𝑣) → ¬ ∀𝑥𝑥 = 𝑦) )
| | 21:20: | ⊢ ( 𝑢 ≠ 𝑣 ▶ ∀𝑦(∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦) )
| | 22:21: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑦∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦) )
| | 23:: | ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ ∃
𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
| | 24:22,23: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦) )
| | 25:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦
)
| | 26:25: | ⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∃𝑦∀𝑦¬
∀𝑥𝑥 = 𝑦)
| | 260:: | ⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦∀𝑦¬
∀𝑥𝑥 = 𝑦)
| | 27:260: | ⊢ (∃𝑦∀𝑦¬ ∀𝑥𝑥 = 𝑦 ↔ ∀𝑦¬
∀𝑥𝑥 = 𝑦)
| | 270:26,27: | ⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥
𝑥 = 𝑦)
| | 28:: | ⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦
)
| | 29:270,28: | ⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦
)
| | 30:24,29: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦) )
| | 31:30: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)) )
| | 32:31: | ⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦
= 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
| | 33:: | ⊢ ( 𝑢 = 𝑣 ▶ 𝑢 = 𝑣 )
| | 34:33: | ⊢ ( 𝑢 = 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦
= 𝑣) → 𝑢 = 𝑣) )
| | 35:34: | ⊢ ( 𝑢 = 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦
= 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)) )
| | 36:35: | ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 =
𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
| | 37:: | ⊢ (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣)
| | 38:32,36,37: | ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (
¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
| | 39:: | ⊢ (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| | 40:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢
∧ 𝑦 = 𝑣))
| | 41:40: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃
𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| | 42:: | ⊢ (∀𝑥𝑥 = 𝑦 ∨ ¬ ∀𝑥𝑥 = 𝑦)
| | 43:39,41,42: | ⊢ (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣
))
| | 44:40,43: | ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ∃𝑥
∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
| | qed:38,44: | ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥
∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
|
|
| ⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
| |
| Theorem | 2sb5ndVD 44882* |
The following User's Proof is a Virtual Deduction proof (see wvd1 44542)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2sb5nd 44533 is 2sb5ndVD 44882 without virtual
deductions and was automatically derived from 2sb5ndVD 44882.
(Contributed by Alan Sare, 30-Apr-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 2:1: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 /
𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 3:: | ⊢ ([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
| | 4:3: | ⊢ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣
/ 𝑦]𝜑)
| | 5:4: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥]
∀𝑦[𝑣 / 𝑦]𝜑)
| | 6:: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ¬ ∀𝑥𝑥 = 𝑦 )
| | 7:: | ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑥𝑥 = 𝑦)
| | 8:7: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑦𝑦 = 𝑥)
| | 9:6,8: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ¬ ∀𝑦𝑦 = 𝑥 )
| | 10:9: | ⊢ ([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀
𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
| | 11:5,10: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢
/ 𝑥][𝑣 / 𝑦]𝜑)
| | 12:11: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 /
𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| | 13:: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑥[𝑢
/ 𝑥][𝑣 / 𝑦]𝜑)
| | 14:: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥𝑥 = 𝑦 )
| | 15:14: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ (∀𝑥[𝑢 / 𝑥][
𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) )
| | 16:13,15: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦
]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) )
| | 17:16: | ⊢ (∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]
𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| | 19:12,17: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢
/ 𝑥][𝑣 / 𝑦]𝜑)
| | 20:19: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 /
𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| | 21:2,20: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)
↔ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| | 22:21: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
𝜑) ↔ ∃𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| | 23:13: | ⊢ (∃𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [
𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| | 24:22,23: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [
𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 240:24: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔
(∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
𝜑)))
| | 241:: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔
(∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| | 242:241,240: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [
𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
| | 243:: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
𝜑))) ↔ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))))
| | 25:242,243: | ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ([
𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
| | 26:: | ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥
∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
| | qed:25,26: | ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢
/ 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
|
|
| ⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) |
| |
| Theorem | 2uasbanhVD 44883* |
The following User's Proof is a Virtual Deduction proof (see wvd1 44542)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2uasbanh 44534 is 2uasbanhVD 44883 without
virtual deductions and was automatically derived from 2uasbanhVD 44883.
(Contributed by Alan Sare, 31-May-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| h1:: | ⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
| | 100:1: | ⊢ (𝜒 → (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
| | 2:100: | ⊢ ( 𝜒 ▶ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦
= 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) )
| | 3:2: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜑) )
| | 4:3: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣
) )
| | 5:4: | ⊢ ( 𝜒 ▶ (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)
)
| | 6:5: | ⊢ ( 𝜒 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑
↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) )
| | 7:3,6: | ⊢ ( 𝜒 ▶ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑 )
| | 8:2: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜓) )
| | 9:5: | ⊢ ( 𝜒 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜓
↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) )
| | 10:8,9: | ⊢ ( 𝜒 ▶ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓 )
| | 101:: | ⊢ ([𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑣 /
𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
| | 102:101: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓)
↔ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
| | 103:: | ⊢ ([𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦
]𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
| | 104:102,103: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓)
↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
| | 11:7,10,104: | ⊢ ( 𝜒 ▶ [𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧
𝜓) )
| | 110:5: | ⊢ ( 𝜒 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑
∧ 𝜓) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓))) )
| | 12:11,110: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ (𝜑 ∧ 𝜓)) )
| | 120:12: | ⊢ (𝜒 → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣
) ∧ (𝜑 ∧ 𝜓)))
| | 13:1,120: | ⊢ ((∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) →
∃𝑥∃𝑦((𝑥 = 𝑢
∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)))
| | 14:: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) )
| | 15:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) )
| | 16:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ (𝜑 ∧ 𝜓) )
| | 17:16: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ 𝜑 )
| | 18:15,17: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) )
| | 19:18: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 20:19: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑
∧ 𝜓)) → ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 21:20: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 22:16: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ 𝜓 )
| | 23:15,22: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓) )
| | 24:23: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
| | 25:24: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑
∧ 𝜓)) → ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
| | 26:25: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
| | 27:21,26: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) → (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧
∃𝑥∃𝑦(
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
| | qed:13,27: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧
∃𝑥∃𝑦(
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
|
|
| ⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) ⇒ ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) |
| |
| Theorem | e2ebindVD 44884 |
The following User's Proof is a Virtual Deduction proof (see wvd1 44542)
completed automatically by a Metamath tools program invoking mmj2 and the
Metamath Proof Assistant. e2ebind 44536 is e2ebindVD 44884 without virtual
deductions and was automatically derived from e2ebindVD 44884.
| 1:: | ⊢ (𝜑 ↔ 𝜑)
| | 2:1: | ⊢ (∀𝑦𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
| | 3:2: | ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑
))
| | 4:: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ ∀𝑦𝑦 = 𝑥 )
| | 5:3,4: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦𝜑 ↔ ∃𝑥
𝜑) )
| | 6:: | ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑦∀𝑦𝑦 = 𝑥)
| | 7:5,6: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ ∀𝑦(∃𝑦𝜑 ↔
∃𝑥𝜑) )
| | 8:7: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦∃𝑦𝜑 ↔
∃𝑦∃𝑥𝜑) )
| | 9:: | ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
| | 10:8,9: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦∃𝑦𝜑 ↔
∃𝑥∃𝑦𝜑) )
| | 11:: | ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
| | 12:11: | ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑)
| | 13:10,12: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑥∃𝑦𝜑 ↔
∃𝑦𝜑) )
| | 14:13: | ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃
𝑦𝜑))
| | 15:: | ⊢ (∀𝑦𝑦 = 𝑥 ↔ ∀𝑥𝑥 = 𝑦)
| | qed:14,15: | ⊢ (∀𝑥𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃
𝑦𝜑))
|
(Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
| |
| 21.41.8 Virtual Deduction transcriptions of
textbook proofs
|
| |
| Theorem | sb5ALTVD 44885* |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Unit 20
Excercise 3.a., which is sb5 2276, found in the "Answers to Starred
Exercises" on page 457 of "Understanding Symbolic Logic", Fifth
Edition (2008), by Virginia Klenk. The same proof may also be
interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It
was completed automatically by the tools program completeusersproof.cmd,
which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. sb5ALT 44498 is sb5ALTVD 44885 without virtual deductions and
was automatically derived from sb5ALTVD 44885.
| 1:: | ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥]𝜑 )
| | 2:: | ⊢ [𝑦 / 𝑥]𝑥 = 𝑦
| | 3:1,2: | ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥](𝑥 = 𝑦
∧ 𝜑) )
| | 4:3: | ⊢ ( [𝑦 / 𝑥]𝜑 ▶ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑
) )
| | 5:4: | ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
)
| | 6:: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ▶ ∃𝑥(𝑥 =
𝑦 ∧ 𝜑) )
| | 7:: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ (𝑥 = 𝑦 ∧ 𝜑) )
| | 8:7: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ 𝜑 )
| | 9:7: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ 𝑥 = 𝑦 )
| | 10:8,9: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ [𝑦 / 𝑥]𝜑 )
| | 101:: | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
| | 11:101,10: | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑
)
| | 12:5,11: | ⊢ (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑
)) ∧ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))
| | qed:12: | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
)
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
| |
| Theorem | vk15.4jVD 44886 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Unit 15
Excercise 4.f. found in the "Answers to Starred Exercises" on page 442
of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia
Klenk. The same proof may also be interpreted to be a Virtual Deduction
Hilbert-style axiomatic proof. It was completed automatically by the
tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant. vk15.4j 44501 is vk15.4jVD 44886
without virtual deductions and was automatically derived
from vk15.4jVD 44886. Step numbers greater than 25 are additional steps
necessary for the sequent calculus proof not contained in the
Fitch-style proof. Otherwise, step i of the User's Proof corresponds to
step i of the Fitch-style proof.
| h1:: | ⊢ ¬ (∃𝑥¬ 𝜑 ∧ ∃𝑥(𝜓 ∧
¬ 𝜒))
| | h2:: | ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏
))
| | h3:: | ⊢ ¬ ∀𝑥(𝜏 → 𝜑)
| | 4:: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∃𝑥¬
𝜃 )
| | 5:4: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∀𝑥𝜃 )
| | 6:3: | ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
| | 7:: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ (𝜏 ∧ ¬ 𝜑) )
| | 8:7: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ 𝜏 )
| | 9:7: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ¬ 𝜑 )
| | 10:5: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ 𝜃 )
| | 11:10,8: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ (𝜃 ∧ 𝜏) )
| | 12:11: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ∃𝑥(𝜃 ∧ 𝜏) )
| | 13:12: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏) )
| | 14:2,13: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ¬ ∀𝑥𝜒 )
| | 140:: | ⊢ (∃𝑥¬ 𝜃 → ∀𝑥∃𝑥¬ 𝜃
)
| | 141:140: | ⊢ (¬ ∃𝑥¬ 𝜃 → ∀𝑥¬ ∃𝑥
¬ 𝜃)
| | 142:: | ⊢ (∀𝑥𝜒 → ∀𝑥∀𝑥𝜒)
| | 143:142: | ⊢ (¬ ∀𝑥𝜒 → ∀𝑥¬ ∀𝑥𝜒
)
| | 144:6,14,141,143: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∀𝑥𝜒
)
| | 15:1: | ⊢ (¬ ∃𝑥¬ 𝜑 ∨ ¬ ∃𝑥(𝜓
∧ ¬ 𝜒))
| | 16:9: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ∃𝑥¬ 𝜑 )
| | 161:: | ⊢ (∃𝑥¬ 𝜑 → ∀𝑥∃𝑥¬ 𝜑
)
| | 162:6,16,141,161: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∃𝑥¬ 𝜑
)
| | 17:162: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ¬ ∃𝑥
¬ 𝜑 )
| | 18:15,17: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∃𝑥(
𝜓 ∧ ¬ 𝜒) )
| | 19:18: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∀𝑥(𝜓
→ 𝜒) )
| | 20:144: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∃𝑥¬ 𝜒
)
| | 21:: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , ¬ 𝜒 ▶ ¬
𝜒 )
| | 22:19: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ (𝜓 → 𝜒
) )
| | 23:21,22: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , ¬ 𝜒 ▶ ¬
𝜓 )
| | 24:23: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , ¬ 𝜒 ▶ ∃
𝑥¬ 𝜓 )
| | 240:: | ⊢ (∃𝑥¬ 𝜓 → ∀𝑥∃𝑥¬ 𝜓
)
| | 241:20,24,141,240: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∃𝑥¬ 𝜓
)
| | 25:241: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∀𝑥𝜓
)
| | qed:25: | ⊢ (¬ ∃𝑥¬ 𝜃 → ¬ ∀𝑥𝜓)
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ¬
(∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) & ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏)) & ⊢ ¬
∀𝑥(𝜏 → 𝜑) ⇒ ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓) |
| |
| Theorem | notnotrALTVD 44887 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Theorem 5 of
Section 14 of [Margaris] p. 59 (which is notnotr 130). The same proof
may also be interpreted as a Virtual Deduction Hilbert-style
axiomatic proof. It was completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. notnotrALT 44502 is notnotrALTVD 44887
without virtual deductions and was automatically derived
from notnotrALTVD 44887. Step i of the User's Proof corresponds to
step i of the Fitch-style proof.
| 1:: | ⊢ ( ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 )
| | 2:: | ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
| | 3:1: | ⊢ ( ¬ ¬ 𝜑 ▶ (¬ 𝜑 → ¬ ¬ ¬ 𝜑) )
| | 4:: | ⊢ ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑 →
𝜑))
| | 5:3: | ⊢ ( ¬ ¬ 𝜑 ▶ (¬ ¬ 𝜑 → 𝜑) )
| | 6:5,1: | ⊢ ( ¬ ¬ 𝜑 ▶ 𝜑 )
| | qed:6: | ⊢ (¬ ¬ 𝜑 → 𝜑)
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (¬ ¬
𝜑 → 𝜑) |
| |
| Theorem | con3ALTVD 44888 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Theorem 7 of
Section 14 of [Margaris] p. 60 (which is con3 153). The same proof may
also be interpreted to be a Virtual Deduction Hilbert-style axiomatic
proof. It was completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. con3ALT2 44503 is con3ALTVD 44888 without
virtual deductions and was automatically derived from con3ALTVD 44888.
Step i of the User's Proof corresponds to step i of the Fitch-style proof.
| 1:: | ⊢ ( (𝜑 → 𝜓) ▶ (𝜑 → 𝜓) )
| | 2:: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 )
| | 3:: | ⊢ (¬ ¬ 𝜑 → 𝜑)
| | 4:2: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜑 )
| | 5:1,4: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜓 )
| | 6:: | ⊢ (𝜓 → ¬ ¬ 𝜓)
| | 7:6,5: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜓 )
| | 8:7: | ⊢ ( (𝜑 → 𝜓) ▶ (¬ ¬ 𝜑 → ¬ ¬ 𝜓
) )
| | 9:: | ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 →
¬ 𝜑))
| | 10:8: | ⊢ ( (𝜑 → 𝜓) ▶ (¬ 𝜓 → ¬ 𝜑) )
| | qed:10: | ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) |
| |
| 21.41.9 Theorems proved using conjunction-form
Virtual Deduction
|
| |
| Theorem | elpwgdedVD 44889 |
Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived
from elpwg 4578. In form of VD deduction with 𝜑 and 𝜓 as
variable virtual hypothesis collections based on Mario Carneiro's
metavariable concept. elpwgded 44537 is elpwgdedVD 44889 using conventional
notation. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ( 𝜑 ▶ 𝐴 ∈ V ) & ⊢ ( 𝜓 ▶ 𝐴 ⊆ 𝐵 )
⇒ ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝐴 ∈ 𝒫 𝐵 ) |
| |
| Theorem | sspwimp 44890 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. For the biconditional, see
sspwb 5424. The proof sspwimp 44890, using conventional notation, was
translated from virtual deduction form, sspwimpVD 44891, using a
translation program. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | sspwimpVD 44891 |
The following User's Proof is a Virtual Deduction proof (see wvd1 44542)
using conjunction-form virtual hypothesis collections. It was completed
manually, but has the potential to be completed automatically by a tools
program which would invoke Mel L. O'Cat's mmj2 and Norm Megill's
Metamath Proof Assistant.
sspwimp 44890 is sspwimpVD 44891 without virtual deductions and was derived
from sspwimpVD 44891. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝐴 ⊆ 𝐵 )
| | 2:: | ⊢ ( .............. 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ∈ 𝒫 𝐴 )
| | 3:2: | ⊢ ( .............. 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ⊆ 𝐴 )
| | 4:3,1: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ⊆ 𝐵 )
| | 5:: | ⊢ 𝑥 ∈ V
| | 6:4,5: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ∈ 𝒫 𝐵
)
| | 7:6: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)
)
| | 8:7: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈
𝒫 𝐵) )
| | 9:8: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝒫 𝐴 ⊆ 𝒫 𝐵 )
| | qed:9: | ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
|
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | sspwimpcf 44892 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. sspwimpcf 44892, using
conventional notation, was translated from its virtual deduction form,
sspwimpcfVD 44893, using a translation program. (Contributed
by Alan Sare,
13-Jun-2015.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | sspwimpcfVD 44893 |
The following User's Proof is a Virtual Deduction proof (see wvd1 44542)
using conjunction-form virtual hypothesis collections. It was completed
automatically by a tools program which would invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant.
sspwimpcf 44892 is sspwimpcfVD 44893 without virtual deductions and was derived
from sspwimpcfVD 44893.
The version of completeusersproof.cmd used is capable of only generating
conjunction-form unification theorems, not unification deductions.
(Contributed by Alan Sare, 13-Jun-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝐴 ⊆ 𝐵 )
| | 2:: | ⊢ ( ........... 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ∈ 𝒫 𝐴 )
| | 3:2: | ⊢ ( ........... 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ⊆ 𝐴 )
| | 4:3,1: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ⊆ 𝐵 )
| | 5:: | ⊢ 𝑥 ∈ V
| | 6:4,5: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ∈ 𝒫 𝐵
)
| | 7:6: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)
)
| | 8:7: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈
𝒫 𝐵) )
| | 9:8: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝒫 𝐴 ⊆ 𝒫 𝐵 )
| | qed:9: | ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
|
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | suctrALTcf 44894 |
The successor of a transitive class is transitive. suctrALTcf 44894, using
conventional notation, was translated from virtual deduction form,
suctrALTcfVD 44895, using a translation program. (Contributed
by Alan
Sare, 13-Jun-2015.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | suctrALTcfVD 44895 |
The following User's Proof is a Virtual Deduction proof (see wvd1 44542)
using conjunction-form virtual hypothesis collections. The
conjunction-form version of completeusersproof.cmd. It allows the User
to avoid superflous virtual hypotheses. This proof was completed
automatically by a tools program which invokes Mel L. O'Cat's
mmj2 and Norm Megill's Metamath Proof Assistant. suctrALTcf 44894
is suctrALTcfVD 44895 without virtual deductions and was derived
automatically from suctrALTcfVD 44895. The version of
completeusersproof.cmd used is capable of only generating
conjunction-form unification theorems, not unification deductions.
(Contributed by Alan Sare, 13-Jun-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( Tr 𝐴 ▶ Tr 𝐴 )
| | 2:: | ⊢ ( ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) )
| | 3:2: | ⊢ ( ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ 𝑧 ∈ 𝑦 )
| | 4:: | ⊢ ( ...................................
....... 𝑦 ∈ 𝐴 ▶ 𝑦 ∈ 𝐴 )
| | 5:1,3,4: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
, 𝑦 ∈ 𝐴 ) ▶ 𝑧 ∈ 𝐴 )
| | 6:: | ⊢ 𝐴 ⊆ suc 𝐴
| | 7:5,6: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
, 𝑦 ∈ 𝐴 ) ▶ 𝑧 ∈ suc 𝐴 )
| | 8:7: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
) ▶ (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) )
| | 9:: | ⊢ ( ...................................
...... 𝑦 = 𝐴 ▶ 𝑦 = 𝐴 )
| | 10:3,9: | ⊢ ( ........ ( (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴), 𝑦 = 𝐴 ) ▶ 𝑧 ∈ 𝐴 )
| | 11:10,6: | ⊢ ( ........ ( (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴), 𝑦 = 𝐴 ) ▶ 𝑧 ∈ suc 𝐴 )
| | 12:11: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) )
| | 13:2: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ 𝑦 ∈ suc 𝐴 )
| | 14:13: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) )
| | 15:8,12,14: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
) ▶ 𝑧 ∈ suc 𝐴 )
| | 16:15: | ⊢ ( Tr 𝐴 ▶ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) → 𝑧 ∈ suc 𝐴) )
| | 17:16: | ⊢ ( Tr 𝐴 ▶ ∀𝑧∀𝑦((𝑧 ∈
𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) )
| | 18:17: | ⊢ ( Tr 𝐴 ▶ Tr suc 𝐴 )
| | qed:18: | ⊢ (Tr 𝐴 → Tr suc 𝐴)
|
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| 21.41.10 Theorems with a VD proof in
conventional notation derived from a VD proof
|
| |
| Theorem | suctrALT3 44896 |
The successor of a transitive class is transitive. suctrALT3 44896 is the
completed proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/suctralt3vd.html 44896.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 44542 for a description of Virtual
Deduction.
Some sub-theorems of the proof were completed using a unification
deduction (e.g., the sub-theorem whose assertion is step 19 used
jaoded 44539). Unification deductions employ Mario
Carneiro's metavariable
concept. Some sub-theorems were completed using a unification theorem
(e.g., the sub-theorem whose assertion is step 24 used dftr2 5231) .
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | sspwimpALT 44897 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. sspwimpALT 44897 is the completed
proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html 44897.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 44542 for a description of Virtual
Deduction.
Some sub-theorems of the proof were completed using a unification
deduction (e.g., the sub-theorem whose assertion is step 9 used
elpwgded 44537). Unification deductions employ Mario
Carneiro's
metavariable concept. Some sub-theorems were completed using a
unification theorem (e.g., the sub-theorem whose assertion is step 5
used elpwi 4582). (Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | unisnALT 44898 |
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
The User manually input on a mmj2 Proof Worksheet, without labels, all
steps of unisnALT 44898 except 1, 11, 15, 21, and 30. With
execution of the
mmj2 unification command, mmj2 could find labels for all steps except
for 2, 12, 16, 22, and 31 (and the then non-existing steps 1, 11, 15,
21, and 30). mmj2 could not find reference theorems for those five steps
because the hypothesis field of each of these steps was empty and none
of those steps unifies with a theorem in set.mm. Each of these five
steps is a semantic variation of a theorem in set.mm and is 2-step
provable. mmj2 does not have the ability to automatically generate the
semantic variation in set.mm of a theorem in a mmj2 Proof Worksheet
unless the theorem in the Proof Worksheet is labeled with a 1-hypothesis
deduction whose hypothesis is a theorem in set.mm which unifies with the
theorem in the Proof Worksheet. The stepprover.c program, which invokes
mmj2, has this capability. stepprover.c automatically generated steps 1,
11, 15, 21, and 30, labeled all steps, and generated the RPN proof of
unisnALT 44898. Roughly speaking, stepprover.c added to
the Proof
Worksheet a labeled duplicate step of each non-unifying theorem for each
label in a text file, labels.txt, containing a list of labels provided
by the User. Upon mmj2 unification, stepprover.c identified a label for
each of the five theorems which 2-step proves it. For unisnALT 44898, the
label list is a list of all 1-hypothesis propositional calculus
deductions in set.mm. stepproverp.c is the same as stepprover.c except
that it intermittently pauses during execution, allowing the User to
observe the changes to a text file caused by the execution of particular
statements of the program. (Contributed by Alan Sare, 19-Aug-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ∪
{𝐴} = 𝐴 |
| |
| 21.41.11 Theorems with a proof in conventional
notation derived from a VD proof
Theorems with a proof in conventional notation automatically derived by
completeusersproof.c from a Virtual Deduction User's Proof.
|
| |
| Theorem | notnotrALT2 44899 |
Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102.
Proof derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. (Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (¬ ¬
𝜑 → 𝜑) |
| |
| Theorem | sspwimpALT2 44900 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. Proof derived by
completeusersproof.c from User's Proof in VirtualDeductionProofs.txt.
The User's Proof in html format is displayed in
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html.
(Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
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