Theorem List for Metamath Proof Explorer - 46001-46100 *Has distinct variable
group(s)
| Type | Label | Description |
|---|
| Statement |
|---|
| |
| Theorem | snssiALT 46001 |
If a class is an element of another class, then its singleton is a
subclass of that other class. Alternate proof of snssi 4746. This
theorem was automatically generated from snssiALTVD 46000 using a
translation program. (Contributed by Alan Sare, 11-Sep-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
| |
| Theorem | snsslVD 46002 |
Virtual deduction proof of snssl 46003. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
| |
| Theorem | snssl 46003 |
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 4745.
The proof of
this theorem was automatically generated from snsslVD 46002 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 46004 |
Virtual deduction proof of snelpwi 5416. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
| |
| Theorem | unipwrVD 46005 |
Virtual deduction proof of unipwr 46006. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
| |
| Theorem | unipwr 46006 |
A class is a subclass of the union of its power class. This theorem is
the right-to-left subclass lemma of unipw 5422. The proof of this theorem
was automatically generated from unipwrVD 46005 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 46007 |
Virtual deduction proof of sstrALT2 46008. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
| |
| Theorem | sstrALT2 46008 |
Virtual deduction proof of sstr 3921, transitivity of subclasses, Theorem
6 of [Suppes] p. 23. This theorem was
automatically generated from
sstrALT2VD 46007 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 46009 |
Virtual deduction proof of suctrALT2 46010. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | suctrALT2 46010 |
Virtual deduction proof of suctr 6469. The successor of a transitive
class is transitive. This proof was generated automatically from the
virtual deduction proof suctrALT2VD 46009 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 46011* |
Virtual deduction proof of elex2 2841. (Contributed by Alan Sare,
25-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
| |
| Theorem | elex22VD 46012* |
Virtual deduction proof of elex22 3478. (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 46013* |
Virtual deduction proof of eqsbc2 3805. (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 46014* |
Virtual deduction proof of zfregs2 9761. (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 46015 |
Virtual deduction proof of tpid3g 4733. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
| |
| Theorem | en3lplem1VD 46016* |
Virtual deduction proof of en3lplem1 9640. (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 46017* |
Virtual deduction proof of en3lplem2 9641. (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 46018 |
Virtual deduction proof of en3lp 9642. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) |
| |
| 21.45.7 Theorems proved using Virtual Deduction
with mmj2 assistance
|
| |
| Theorem | simplbi2VD 46019 |
Virtual deduction proof of simplbi2 506. 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 45945 | ⊢ ((𝜓 ∧ 𝜒) → 𝜑)
| | qed:3,?: e0a 45945 | ⊢ (𝜓 → (𝜒 → 𝜑))
|
The proof of simplbi2 506 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 46020 |
Virtual deduction proof of 3ornot23 45683. 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 45801 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜑 )
| | 4:1,?: e1a 45801 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜓 )
| | 5:3,4,?: e11 45862 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ (𝜑
∨ 𝜓) )
| | 6:2,?: e2 45805 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ (𝜒 ∨ (𝜑 ∨ 𝜓)) )
| | 7:5,6,?: e12 45897 | ⊢ ( (¬ 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 46021 |
Virtual deduction proof of orbi1r 45684. 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 45805 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜑 ∨ 𝜒) )
| | 4:1,3,?: e12 45897 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜓 ∨ 𝜒) )
| | 5:4,?: e2 45805 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜒 ∨ 𝜓) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜑)
→ (𝜒 ∨ 𝜓)) )
| | 7:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜓) )
| | 8:7,?: e2 45805 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜓 ∨ 𝜒) )
| | 9:1,8,?: e12 45897 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜑 ∨ 𝜒) )
| | 10:9,?: e2 45805 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜑) )
| | 11:10: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜓)
→ (𝜒 ∨ 𝜑)) )
| | 12:6,11,?: e11 45862 | ⊢ ( (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 46022 |
Virtual deduction proof of bitr3 355. 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 45801 | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜓
↔ 𝜑) )
| | 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜑 ↔ 𝜒) )
| | 4:3,?: e2 45805 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜒 ↔ 𝜑) )
| | 5:2,4,?: e12 45897 | ⊢ ( (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 46023 |
Virtual deduction proof of 3orbi123 45685. 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 45801 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜑 ↔ 𝜓) )
| | 3:1,?: e1a 45801 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜒 ↔ 𝜃) )
| | 4:1,?: e1a 45801 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜏 ↔ 𝜂) )
| | 5:2,3,?: e11 45862 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) )
| | 6:5,4,?: e11 45862 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 7:?: | ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ (𝜑
∨ 𝜒 ∨ 𝜏))
| | 8:6,7,?: e10 45868 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 9:?: | ⊢ (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔
(𝜓 ∨ 𝜃 ∨ 𝜂))
| | 10:8,9,?: e10 45868 | ⊢ ( ((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 46024 |
Virtual deduction proof of the analogue of sbcor 3792 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 45801 | ⊢ ( 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 45868 | ⊢ ( 𝐴 ∈ 𝐵 ▶ [𝐴 / 𝑥](((𝜑
∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 4:1,33,?: e11 45862 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝜑
∨ 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 5:2,4,?: e11 45862 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒)) )
| | 6:1,?: e1a 45801 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) )
| | 7:6,?: e1a 45801 | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥](𝜑
∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 8:5,7,?: e11 45862 | ⊢ ( 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 45868 | ⊢ ( 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 46025* |
Virtual deduction proof of alrim3con13v 45707. 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 45805 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜓 )
| | 4:2,?: e2 45805 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜑 )
| | 5:2,?: e2 45805 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜒 )
| | 6:1,4,?: e12 45897 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜑 )
| | 7:3,?: e2 45805 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜓 )
| | 8:5,?: e2 45805 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜒 )
| | 9:7,6,8,?: e222 45810 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒) )
| | 10:9,?: e2 45805 | ⊢ ( (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 46026 |
Virtual deduction proof of exbir 45653. 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 45897 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓) ▶ (𝜒 ↔ 𝜃) )
| | 6:3,5,?: e32 45931 | ⊢ ( ((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 45801 | ⊢ ( ((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 46027 |
Virtual deduction proof of exbiri 823. 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 45868 | ⊢ ( 𝜑 ▶ (𝜓 → (𝜒 ↔ 𝜃)) )
| | 6:3,5,?: e21 45903 | ⊢ ( 𝜑 , 𝜓 ▶ (𝜒 ↔ 𝜃) )
| | 7:4,6,?: e32 45931 | ⊢ ( 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 46028* |
Virtual deduction proof of rspsbc2 45708. 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 45921 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑 )
| | 5:1,4,?: e13 45921 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑 )
| | 6:2,5,?: e23 45928 | ⊢ ( 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 46029 |
Virtual deduction proof of 3impexp 1377. 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 45868 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 4:3,?: e1a 45801 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 5:4,?: e1a 45801 | ⊢ ( ((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 45801 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 9:8,?: e1a 45801 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 10:2,9,?: e01 45865 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) )
| | 11:10: | ⊢ ((𝜑 → (𝜓 → (𝜒
→ 𝜃))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
| | qed:6,11,?: e00 45941 | ⊢ (((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 46030 |
Virtual deduction proof of 3impexpbicom 45654. 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 45868 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃)) )
| | 4:3,?: e1a 45801 | ⊢ ( ((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 45801 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏
↔ 𝜃)) )
| | 8:7,2,?: e10 45868 | ⊢ ( (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 45941 | ⊢ (((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 46031 |
Virtual deduction proof of 3impexpbicomi 45655. 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 45945 | ⊢ (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 46032* |
Virtual deduction proof of sbcoreleleq 45709. 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 45801 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 ∈
𝑦 ↔ 𝑥 ∈ 𝐴) )
| | 3:1,?: e1a 45801 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑦 ∈
𝑥 ↔ 𝐴 ∈ 𝑥) )
| | 4:1,?: e1a 45801 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 =
𝑦 ↔ 𝑥 = 𝐴) )
| | 5:2,3,4,?: e111 45848 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ((𝑥 ∈ 𝐴
∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥
∨ [𝐴 / 𝑦]𝑥 = 𝑦)) )
| | 6:1,?: e1a 45801 | ⊢ ( 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 45862 | ⊢ ( 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 46033* |
Virtual deduction proof of nfra2 3365. 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 45941 | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 →
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 4:2: | ⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 5:4,?: e0a 45945 | ⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | qed:3,5,?: e00 45941 | ⊢ (∀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 46034* |
Virtual deduction proof of tratrb 45710. 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 45801 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| | 3:1,?: e1a 45801 | ⊢ ( (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 45801 | ⊢ ( (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 45805 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝑦 )
| | 7:5,?: e2 45805 | ⊢ ( (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 45830 | ⊢ ( (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 45827 | ⊢ ( (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 45809 | ⊢ ( (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 45900 | ⊢ ( (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 45928 | ⊢ ( (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 45928 | ⊢ ( (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 45900 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝑥 = 𝐵 )
| | 21:3,?: e1a 45801 | ⊢ ( (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 45830 | ⊢ ( (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 45805 | ⊢ ( (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 45897 | ⊢ ( (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 45810 | ⊢ ( (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 45945 | ⊢ ((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 45945 | ⊢ ((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 45801 | ⊢ ( (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 46035 |
Virtual deduction proof of al2im 1847. 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 45801 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)) )
| | 3:: | ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓
→ ∀𝑥𝜒))
| | 4:2,3,?: e10 45868 | ⊢ ( ∀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 46036 |
Virtual deduction proof of syl5imp 45686. 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 45801 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜓
→ (𝜑 → 𝜒)) )
| | 3:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → 𝜓) )
| | 4:3,2,?: e21 45903 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → (𝜑 → 𝜒)) )
| | 5:4,?: e2 45805 | ⊢ ( (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 46037 |
Virtual deduction proof of idiALT 45652. 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 45945 | ⊢ 𝜑
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝜑 ⇒ ⊢ 𝜑 |
| |
| Theorem | ancomstVD 46038 |
Closed form of ancoms 464. 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 45945 | ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓
∧ 𝜑) → 𝜒))
|
The proof of ancomst 470 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 46039* |
Quantification restricted to a subclass for two quantifiers. ssralv 3982
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 45705 is ssralv2VD 46039 without
virtual deductions and was automatically derived from ssralv2VD 46039.
| 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 46040 |
An element of an ordinal class is ordinal. Proposition 7.6 of
[TakeutiZaring] p. 36. This is an alternate proof of ordelord 6402 using
the Axiom of Regularity indirectly through dford2 9648. 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 45711 is ordelordALTVD 46040
without virtual deductions and was automatically derived from
ordelordALTVD 46040 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 46041 |
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 4161 is equncomVD 46041 without
virtual deductions and was automatically derived from equncomVD 46041.
| 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 46042 |
Inference form of equncom 4161. 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 4162 is equncomiVD 46042 without
virtual deductions and was automatically derived from equncomiVD 46042.
| 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 46043 |
A set belongs to its successor. Alternate proof of sucid 6465.
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 46044 is sucidALTVD 46043
without virtual deductions and was automatically derived from
sucidALTVD 46043. 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 6386, 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 9648.
| 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 46044 |
A set belongs to its successor. This proof was automatically derived
from sucidALTVD 46043 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 46045 |
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 6465 is sucidVD 46045 without virtual deductions and was automatically
derived from sucidVD 46045.
| 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 46046 |
Implication form of imbi12i 353. 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 349 is imbi12VD 46046 without virtual
deductions and was automatically derived from imbi12VD 46046.
| 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 46047 |
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 45694
is imbi13VD 46047 without virtual deductions and was automatically derived
from imbi13VD 46047.
| 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 46048 |
Distribution of class substitution over a left-nested implication.
Similar to sbcimg 3790.
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 45712 is sbcim2gVD 46048 without virtual deductions and was automatically
derived from sbcim2gVD 46048.
| 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 46049 |
Implication form of sbcbii 3798.
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 45713 is sbcbiVD 46049 without virtual deductions and was automatically
derived from sbcbiVD 46049.
| 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 46050* |
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 45714 is trsbcVD 46050 without virtual deductions and was automatically
derived from trsbcVD 46050.
| 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 46051* |
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 45715 is truniALTVD 46051 without virtual deductions and was
automatically derived from truniALTVD 46051.
| 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 46052 |
Non-virtual deduction form of e33 45907.
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 45695 is ee33VD 46052 without virtual deductions and was automatically
derived from ee33VD 46052.
| 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 46053* |
The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 46054.
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 46054 is trintALTVD 46053 without virtual deductions and was
automatically derived from trintALTVD 46053.
| 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 46054* |
The intersection of a class of transitive sets is transitive. Exercise
5(b) of [Enderton] p. 73. trintALT 46054 is an alternate proof of trint 5233.
trintALT 46054 is trintALTVD 46053 without virtual deductions and was
automatically derived from trintALTVD 46053 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 46055 |
The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual
deduction proof of undif3 4248.
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 4248 is undif3VD 46055 without virtual deductions and was automatically
derived from undif3VD 46055.
| 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 46056 |
Virtual deduction proof of sbcssg 4480.
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 4480 is sbcssgVD 46056 without virtual deductions and was automatically
derived from sbcssgVD 46056.
| 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 46057 |
Virtual deduction proof of csbin 4403.
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 4403 is csbingVD 46057 without virtual deductions and was
automatically derived from csbingVD 46057.
| 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 46058* |
Virtual deduction proof of onfrALTlem5 45716.
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 45716 is onfrALTlem5VD 46058 without virtual deductions and was
automatically derived from onfrALTlem5VD 46058.
| 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 46059* |
Virtual deduction proof of onfrALTlem4 45717.
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 45717 is onfrALTlem4VD 46059 without virtual deductions and was
automatically derived from onfrALTlem4VD 46059.
| 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 46060* |
Virtual deduction proof of onfrALTlem3 45718.
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 45718 is onfrALTlem3VD 46060 without virtual deductions and was
automatically derived from onfrALTlem3VD 46060.
| 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 46061 |
Virtual deduction proof of simplbi2comt 507.
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 507 is simplbi2comtVD 46061 without virtual deductions and was
automatically derived from simplbi2comtVD 46061.
| 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 46062* |
Virtual deduction proof of onfrALTlem2 45720.
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 45720 is onfrALTlem2VD 46062 without virtual deductions and was
automatically derived from onfrALTlem2VD 46062.
| 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 46063* |
Virtual deduction proof of onfrALTlem1 45722.
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 45722 is onfrALTlem1VD 46063 without virtual deductions and was
automatically derived from onfrALTlem1VD 46063.
| 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 46064 |
Virtual deduction proof of onfrALT 45723.
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 45723 is onfrALTVD 46064 without virtual deductions and was
automatically derived from onfrALTVD 46064.
| 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 46065 |
Virtual deduction proof of csbeq2 3856.
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 3856 is csbeq2gVD 46065 without virtual deductions and was
automatically derived from csbeq2gVD 46065.
| 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 46066 |
Virtual deduction proof of csbsng 4669.
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 4669 is csbsngVD 46066 without virtual deductions and was automatically
derived from csbsngVD 46066.
| 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 46067 |
Virtual deduction proof of csbxp 5757.
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 5757 is csbxpgVD 46067 without virtual deductions and was
automatically derived from csbxpgVD 46067.
| 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 46068 |
Virtual deduction proof of csbres 5984.
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 5984 is csbresgVD 46068 without virtual deductions and was
automatically derived from csbresgVD 46068.
| 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 46069 |
Virtual deduction proof of csbrn 6217.
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 6217 is csbrngVD 46069 without virtual deductions and was
automatically derived from csbrngVD 46069.
| 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 46070 |
Virtual deduction proof of csbima12 6088.
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 6088 is csbima12gALTVD 46070 without virtual deductions and was
automatically derived from csbima12gALTVD 46070.
| 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 46071 |
Virtual deduction proof of csbuni 4901.
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 4901 is csbunigVD 46071 without virtual deductions and was
automatically derived from csbunigVD 46071.
| 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 46072 |
Virtual deduction proof of csbfv12 6953.
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 6953 is csbfv12gALTVD 46072 without virtual deductions and was
automatically derived from csbfv12gALTVD 46072.
| 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 46073 |
Virtual deduction proof of con5 45696.
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 45696 is con5VD 46073 without virtual deductions and was automatically
derived from con5VD 46073.
| 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 46074 |
Virtual deduction proof of relopab 5808.
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 5808 is relopabVD 46074 without virtual deductions and was
automatically derived from relopabVD 46074.
| 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 46075 |
Virtual deduction proof of 19.41rg 45724.
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 45724
is 19.41rgVD 46075 without virtual deductions and was automatically derived
from 19.41rgVD 46075. (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 46076 |
Virtual deduction proof of 2pm13.193 45726.
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 45726 is 2pm13.193VD 46076 without virtual deductions and was
automatically derived from 2pm13.193VD 46076. (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 46077 |
Virtual deduction proof of hbimpg 45728.
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 45728
is hbimpgVD 46077 without virtual deductions and was automatically derived
from hbimpgVD 46077. (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 46078 |
Virtual deduction proof of hbalg 45729.
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 45729
is hbalgVD 46078 without virtual deductions and was automatically derived
from hbalgVD 46078. (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 46079 |
Virtual deduction proof of hbexg 45730.
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 45730
is hbexgVD 46079 without virtual deductions and was automatically derived
from hbexgVD 46079. (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 46080* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45743)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 45731 is ax6e2eqVD 46080 without virtual
deductions and was automatically derived from ax6e2eqVD 46080.
(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 46081* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45743)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2nd 45732 is ax6e2ndVD 46081 without virtual
deductions and was automatically derived from ax6e2ndVD 46081.
(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 46082* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45743)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 45731 is ax6e2ndeqVD 46082 without virtual
deductions and was automatically derived from ax6e2ndeqVD 46082.
(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 46083* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45743)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2sb5nd 45734 is 2sb5ndVD 46083 without virtual
deductions and was automatically derived from 2sb5ndVD 46083.
(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 46084* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45743)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2uasbanh 45735 is 2uasbanhVD 46084 without
virtual deductions and was automatically derived from 2uasbanhVD 46084.
(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 46085 |
The following User's Proof is a Virtual Deduction proof (see wvd1 45743)
completed automatically by a Metamath tools program invoking mmj2 and the
Metamath Proof Assistant. e2ebind 45737 is e2ebindVD 46085 without virtual
deductions and was automatically derived from e2ebindVD 46085.
| 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)) |
| |
| 21.45.8 Virtual Deduction transcriptions of
textbook proofs
|
| |
| Theorem | sb5ALTVD 46086* |
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 2313, 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 45699 is sb5ALTVD 46086 without virtual deductions and
was automatically derived from sb5ALTVD 46086.
| 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 46087 |
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 45702 is vk15.4jVD 46087
without virtual deductions and was automatically derived
from vk15.4jVD 46087. 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 46088 |
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 131). 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 45703 is notnotrALTVD 46088
without virtual deductions and was automatically derived
from notnotrALTVD 46088. 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 46089 |
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 154). 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 45704 is con3ALTVD 46089 without
virtual deductions and was automatically derived from con3ALTVD 46089.
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)) |
| |
| 21.45.9 Theorems proved using conjunction-form
Virtual Deduction
|
| |
| Theorem | elpwgdedVD 46090 |
Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived
from elpwg 4563. In form of VD deduction with 𝜑 and 𝜓 as
variable virtual hypothesis collections based on Mario Carneiro's
metavariable concept. elpwgded 45738 is elpwgdedVD 46090 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 46091 |
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 5421. The proof sspwimp 46091, using conventional notation, was
translated from virtual deduction form, sspwimpVD 46092, 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 46092 |
The following User's Proof is a Virtual Deduction proof (see wvd1 45743)
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 46091 is sspwimpVD 46092 without virtual deductions and was derived
from sspwimpVD 46092. (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 46093 |
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 46093, using
conventional notation, was translated from its virtual deduction form,
sspwimpcfVD 46094, 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 46094 |
The following User's Proof is a Virtual Deduction proof (see wvd1 45743)
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 46093 is sspwimpcfVD 46094 without virtual deductions and was derived
from sspwimpcfVD 46094.
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 46095 |
The successor of a transitive class is transitive. suctrALTcf 46095, using
conventional notation, was translated from virtual deduction form,
suctrALTcfVD 46096, 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 46096 |
The following User's Proof is a Virtual Deduction proof (see wvd1 45743)
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 46095
is suctrALTcfVD 46096 without virtual deductions and was derived
automatically from suctrALTcfVD 46096. 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) |
| |
| 21.45.10 Theorems with a VD proof in
conventional notation derived from a VD proof
|
| |
| Theorem | suctrALT3 46097 |
The successor of a transitive class is transitive. suctrALT3 46097 is the
completed proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/suctralt3vd.html 46097.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 45743 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 45740). 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 5217) .
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | sspwimpALT 46098 |
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 46098 is the completed
proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html 46098.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 45743 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 45738). 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 4567). (Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | unisnALT 46099 |
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 46099 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 46099. 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 46099, 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 |
| |
| 21.45.11 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 46100 |
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) |
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