Theorem List for Metamath Proof Explorer - 45401-45500 *Has distinct variable
group(s)
| Type | Label | Description |
|---|
| Statement |
|---|
| |
| Theorem | snssiALT 45401 |
If a class is an element of another class, then its singleton is a
subclass of that other class. Alternate proof of snssi 4747. This
theorem was automatically generated from snssiALTVD 45400 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 45402 |
Virtual deduction proof of snssl 45403. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
| |
| Theorem | snssl 45403 |
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 4746.
The proof of
this theorem was automatically generated from snsslVD 45402 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 45404 |
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 45405 |
Virtual deduction proof of unipwr 45406. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
| |
| Theorem | unipwr 45406 |
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 45405 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 45407 |
Virtual deduction proof of sstrALT2 45408. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
| |
| Theorem | sstrALT2 45408 |
Virtual deduction proof of sstr 3947, transitivity of subclasses, Theorem
6 of [Suppes] p. 23. This theorem was
automatically generated from
sstrALT2VD 45407 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 45409 |
Virtual deduction proof of suctrALT2 45410. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | suctrALT2 45410 |
Virtual deduction proof of suctr 6438. The successor of a transitive
class is transitive. This proof was generated automatically from the
virtual deduction proof suctrALT2VD 45409 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 45411* |
Virtual deduction proof of elex2 2842. (Contributed by Alan Sare,
25-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
| |
| Theorem | elex22VD 45412* |
Virtual deduction proof of elex22 3481. (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 45413* |
Virtual deduction proof of eqsbc2 3810. (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 45414* |
Virtual deduction proof of zfregs2 9690. (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 45415 |
Virtual deduction proof of tpid3g 4734. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
| |
| Theorem | en3lplem1VD 45416* |
Virtual deduction proof of en3lplem1 9569. (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 45417* |
Virtual deduction proof of en3lplem2 9570. (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 45418 |
Virtual deduction proof of en3lp 9571. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) |
| |
| 21.41.7 Theorems proved using Virtual Deduction
with mmj2 assistance
|
| |
| Theorem | simplbi2VD 45419 |
Virtual deduction proof of simplbi2 505. 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 45345 | ⊢ ((𝜓 ∧ 𝜒) → 𝜑)
| | qed:3,?: e0a 45345 | ⊢ (𝜓 → (𝜒 → 𝜑))
|
The proof of simplbi2 505 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 45420 |
Virtual deduction proof of 3ornot23 45083. 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 45201 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜑 )
| | 4:1,?: e1a 45201 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜓 )
| | 5:3,4,?: e11 45262 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ (𝜑
∨ 𝜓) )
| | 6:2,?: e2 45205 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ (𝜒 ∨ (𝜑 ∨ 𝜓)) )
| | 7:5,6,?: e12 45297 | ⊢ ( (¬ 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 45421 |
Virtual deduction proof of orbi1r 45084. 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 45205 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜑 ∨ 𝜒) )
| | 4:1,3,?: e12 45297 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜓 ∨ 𝜒) )
| | 5:4,?: e2 45205 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜒 ∨ 𝜓) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜑)
→ (𝜒 ∨ 𝜓)) )
| | 7:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜓) )
| | 8:7,?: e2 45205 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜓 ∨ 𝜒) )
| | 9:1,8,?: e12 45297 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜑 ∨ 𝜒) )
| | 10:9,?: e2 45205 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜑) )
| | 11:10: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜓)
→ (𝜒 ∨ 𝜑)) )
| | 12:6,11,?: e11 45262 | ⊢ ( (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 45422 |
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 45201 | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜓
↔ 𝜑) )
| | 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜑 ↔ 𝜒) )
| | 4:3,?: e2 45205 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜒 ↔ 𝜑) )
| | 5:2,4,?: e12 45297 | ⊢ ( (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 45423 |
Virtual deduction proof of 3orbi123 45085. 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 45201 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜑 ↔ 𝜓) )
| | 3:1,?: e1a 45201 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜒 ↔ 𝜃) )
| | 4:1,?: e1a 45201 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜏 ↔ 𝜂) )
| | 5:2,3,?: e11 45262 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) )
| | 6:5,4,?: e11 45262 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 7:?: | ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ (𝜑
∨ 𝜒 ∨ 𝜏))
| | 8:6,7,?: e10 45268 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 9:?: | ⊢ (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔
(𝜓 ∨ 𝜃 ∨ 𝜂))
| | 10:8,9,?: e10 45268 | ⊢ ( ((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 45424 |
Virtual deduction proof of the analogue of sbcor 3797 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 45201 | ⊢ ( 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 45268 | ⊢ ( 𝐴 ∈ 𝐵 ▶ [𝐴 / 𝑥](((𝜑
∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 4:1,33,?: e11 45262 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝜑
∨ 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 5:2,4,?: e11 45262 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒)) )
| | 6:1,?: e1a 45201 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) )
| | 7:6,?: e1a 45201 | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥](𝜑
∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 8:5,7,?: e11 45262 | ⊢ ( 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 45268 | ⊢ ( 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 45425* |
Virtual deduction proof of alrim3con13v 45107. 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 45205 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜓 )
| | 4:2,?: e2 45205 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜑 )
| | 5:2,?: e2 45205 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜒 )
| | 6:1,4,?: e12 45297 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜑 )
| | 7:3,?: e2 45205 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜓 )
| | 8:5,?: e2 45205 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜒 )
| | 9:7,6,8,?: e222 45210 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒) )
| | 10:9,?: e2 45205 | ⊢ ( (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 45426 |
Virtual deduction proof of exbir 45053. 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 45297 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓) ▶ (𝜒 ↔ 𝜃) )
| | 6:3,5,?: e32 45331 | ⊢ ( ((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 45201 | ⊢ ( ((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 45427 |
Virtual deduction proof of exbiri 822. 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 45268 | ⊢ ( 𝜑 ▶ (𝜓 → (𝜒 ↔ 𝜃)) )
| | 6:3,5,?: e21 45303 | ⊢ ( 𝜑 , 𝜓 ▶ (𝜒 ↔ 𝜃) )
| | 7:4,6,?: e32 45331 | ⊢ ( 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 45428* |
Virtual deduction proof of rspsbc2 45108. 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 45321 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑 )
| | 5:1,4,?: e13 45321 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑 )
| | 6:2,5,?: e23 45328 | ⊢ ( 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 45429 |
Virtual deduction proof of 3impexp 1375. 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 45268 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 4:3,?: e1a 45201 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 5:4,?: e1a 45201 | ⊢ ( ((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 45201 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 9:8,?: e1a 45201 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 10:2,9,?: e01 45265 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) )
| | 11:10: | ⊢ ((𝜑 → (𝜓 → (𝜒
→ 𝜃))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
| | qed:6,11,?: e00 45341 | ⊢ (((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 45430 |
Virtual deduction proof of 3impexpbicom 45054. 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 45268 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃)) )
| | 4:3,?: e1a 45201 | ⊢ ( ((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 45201 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏
↔ 𝜃)) )
| | 8:7,2,?: e10 45268 | ⊢ ( (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 45341 | ⊢ (((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 45431 |
Virtual deduction proof of 3impexpbicomi 45055. 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 45345 | ⊢ (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 45432* |
Virtual deduction proof of sbcoreleleq 45109. 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 45201 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 ∈
𝑦 ↔ 𝑥 ∈ 𝐴) )
| | 3:1,?: e1a 45201 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑦 ∈
𝑥 ↔ 𝐴 ∈ 𝑥) )
| | 4:1,?: e1a 45201 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 =
𝑦 ↔ 𝑥 = 𝐴) )
| | 5:2,3,4,?: e111 45248 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ((𝑥 ∈ 𝐴
∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥
∨ [𝐴 / 𝑦]𝑥 = 𝑦)) )
| | 6:1,?: e1a 45201 | ⊢ ( 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 45262 | ⊢ ( 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 45433* |
Virtual deduction proof of nfra2 3366. 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 45341 | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 →
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 4:2: | ⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 5:4,?: e0a 45345 | ⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | qed:3,5,?: e00 45341 | ⊢ (∀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 45434* |
Virtual deduction proof of tratrb 45110. 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 45201 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| | 3:1,?: e1a 45201 | ⊢ ( (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 45201 | ⊢ ( (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 45205 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝑦 )
| | 7:5,?: e2 45205 | ⊢ ( (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 45230 | ⊢ ( (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 45227 | ⊢ ( (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 45209 | ⊢ ( (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 45300 | ⊢ ( (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 45328 | ⊢ ( (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 45328 | ⊢ ( (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 45300 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝑥 = 𝐵 )
| | 21:3,?: e1a 45201 | ⊢ ( (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 45230 | ⊢ ( (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 45205 | ⊢ ( (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 45297 | ⊢ ( (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 45210 | ⊢ ( (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 45345 | ⊢ ((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 45345 | ⊢ ((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 45201 | ⊢ ( (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 45435 |
Virtual deduction proof of al2im 1837. 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 45201 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)) )
| | 3:: | ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓
→ ∀𝑥𝜒))
| | 4:2,3,?: e10 45268 | ⊢ ( ∀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 45436 |
Virtual deduction proof of syl5imp 45086. 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 45201 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜓
→ (𝜑 → 𝜒)) )
| | 3:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → 𝜓) )
| | 4:3,2,?: e21 45303 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → (𝜑 → 𝜒)) )
| | 5:4,?: e2 45205 | ⊢ ( (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 45437 |
Virtual deduction proof of idiALT 45052. 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 45345 | ⊢ 𝜑
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝜑 ⇒ ⊢ 𝜑 |
| |
| Theorem | ancomstVD 45438 |
Closed form of ancoms 463. 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 45345 | ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓
∧ 𝜑) → 𝜒))
|
The proof of ancomst 469 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 45439* |
Quantification restricted to a subclass for two quantifiers. ssralv 4008
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 45105 is ssralv2VD 45439 without
virtual deductions and was automatically derived from ssralv2VD 45439.
| 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 45440 |
An element of an ordinal class is ordinal. Proposition 7.6 of
[TakeutiZaring] p. 36. This is an alternate proof of ordelord 6372 using
the Axiom of Regularity indirectly through dford2 9577. 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 45111 is ordelordALTVD 45440
without virtual deductions and was automatically derived from
ordelordALTVD 45440 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 45441 |
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 4115 is equncomVD 45441 without
virtual deductions and was automatically derived from equncomVD 45441.
| 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 45442 |
Inference form of equncom 4115. 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 4116 is equncomiVD 45442 without
virtual deductions and was automatically derived from equncomiVD 45442.
| 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 45443 |
A set belongs to its successor. Alternate proof of sucid 6434.
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 45444 is sucidALTVD 45443
without virtual deductions and was automatically derived from
sucidALTVD 45443. 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 6356, 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 9577.
| 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 45444 |
A set belongs to its successor. This proof was automatically derived
from sucidALTVD 45443 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 45445 |
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 6434 is sucidVD 45445 without virtual deductions and was automatically
derived from sucidVD 45445.
| 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 45446 |
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 45446 without virtual
deductions and was automatically derived from imbi12VD 45446.
| 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 45447 |
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 45094
is imbi13VD 45447 without virtual deductions and was automatically derived
from imbi13VD 45447.
| 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 45448 |
Distribution of class substitution over a left-nested implication.
Similar to sbcimg 3795.
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 45112 is sbcim2gVD 45448 without virtual deductions and was automatically
derived from sbcim2gVD 45448.
| 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 45449 |
Implication form of sbcbii 3803.
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 45113 is sbcbiVD 45449 without virtual deductions and was automatically
derived from sbcbiVD 45449.
| 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 45450* |
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 45114 is trsbcVD 45450 without virtual deductions and was automatically
derived from trsbcVD 45450.
| 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 45451* |
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 45115 is truniALTVD 45451 without virtual deductions and was
automatically derived from truniALTVD 45451.
| 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 45452 |
Non-virtual deduction form of e33 45307.
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 45095 is ee33VD 45452 without virtual deductions and was automatically
derived from ee33VD 45452.
| 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 45453* |
The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 45454.
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 45454 is trintALTVD 45453 without virtual deductions and was
automatically derived from trintALTVD 45453.
| 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 45454* |
The intersection of a class of transitive sets is transitive. Exercise
5(b) of [Enderton] p. 73. trintALT 45454 is an alternate proof of trint 5230.
trintALT 45454 is trintALTVD 45453 without virtual deductions and was
automatically derived from trintALTVD 45453 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 45455 |
The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual
deduction proof of undif3 4255.
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 4255 is undif3VD 45455 without virtual deductions and was automatically
derived from undif3VD 45455.
| 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 45456 |
Virtual deduction proof of sbcssg 4478.
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 4478 is sbcssgVD 45456 without virtual deductions and was automatically
derived from sbcssgVD 45456.
| 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 45457 |
Virtual deduction proof of csbin 4399.
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 4399 is csbingVD 45457 without virtual deductions and was
automatically derived from csbingVD 45457.
| 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 45458* |
Virtual deduction proof of onfrALTlem5 45116.
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 45116 is onfrALTlem5VD 45458 without virtual deductions and was
automatically derived from onfrALTlem5VD 45458.
| 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 45459* |
Virtual deduction proof of onfrALTlem4 45117.
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 45117 is onfrALTlem4VD 45459 without virtual deductions and was
automatically derived from onfrALTlem4VD 45459.
| 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 45460* |
Virtual deduction proof of onfrALTlem3 45118.
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 45118 is onfrALTlem3VD 45460 without virtual deductions and was
automatically derived from onfrALTlem3VD 45460.
| 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 45461 |
Virtual deduction proof of simplbi2comt 506.
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 506 is simplbi2comtVD 45461 without virtual deductions and was
automatically derived from simplbi2comtVD 45461.
| 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 45462* |
Virtual deduction proof of onfrALTlem2 45120.
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 45120 is onfrALTlem2VD 45462 without virtual deductions and was
automatically derived from onfrALTlem2VD 45462.
| 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 45463* |
Virtual deduction proof of onfrALTlem1 45122.
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 45122 is onfrALTlem1VD 45463 without virtual deductions and was
automatically derived from onfrALTlem1VD 45463.
| 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 45464 |
Virtual deduction proof of onfrALT 45123.
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 45123 is onfrALTVD 45464 without virtual deductions and was
automatically derived from onfrALTVD 45464.
| 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 45465 |
Virtual deduction proof of csbeq2 3860.
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 3860 is csbeq2gVD 45465 without virtual deductions and was
automatically derived from csbeq2gVD 45465.
| 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 45466 |
Virtual deduction proof of csbsng 4670.
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 4670 is csbsngVD 45466 without virtual deductions and was automatically
derived from csbsngVD 45466.
| 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 45467 |
Virtual deduction proof of csbxp 5753.
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 5753 is csbxpgVD 45467 without virtual deductions and was
automatically derived from csbxpgVD 45467.
| 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 45468 |
Virtual deduction proof of csbres 5972.
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 5972 is csbresgVD 45468 without virtual deductions and was
automatically derived from csbresgVD 45468.
| 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 45469 |
Virtual deduction proof of csbrn 6194.
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 6194 is csbrngVD 45469 without virtual deductions and was
automatically derived from csbrngVD 45469.
| 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 45470 |
Virtual deduction proof of csbima12 6072.
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 6072 is csbima12gALTVD 45470 without virtual deductions and was
automatically derived from csbima12gALTVD 45470.
| 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 45471 |
Virtual deduction proof of csbuni 4899.
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 4899 is csbunigVD 45471 without virtual deductions and was
automatically derived from csbunigVD 45471.
| 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 45472 |
Virtual deduction proof of csbfv12 6916.
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 6916 is csbfv12gALTVD 45472 without virtual deductions and was
automatically derived from csbfv12gALTVD 45472.
| 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 45473 |
Virtual deduction proof of con5 45096.
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 45096 is con5VD 45473 without virtual deductions and was automatically
derived from con5VD 45473.
| 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 45474 |
Virtual deduction proof of relopab 5802.
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 5802 is relopabVD 45474 without virtual deductions and was
automatically derived from relopabVD 45474.
| 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 45475 |
Virtual deduction proof of 19.41rg 45124.
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 45124
is 19.41rgVD 45475 without virtual deductions and was automatically derived
from 19.41rgVD 45475. (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 45476 |
Virtual deduction proof of 2pm13.193 45126.
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 45126 is 2pm13.193VD 45476 without virtual deductions and was
automatically derived from 2pm13.193VD 45476. (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 45477 |
Virtual deduction proof of hbimpg 45128.
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 45128
is hbimpgVD 45477 without virtual deductions and was automatically derived
from hbimpgVD 45477. (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 45478 |
Virtual deduction proof of hbalg 45129.
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 45129
is hbalgVD 45478 without virtual deductions and was automatically derived
from hbalgVD 45478. (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 45479 |
Virtual deduction proof of hbexg 45130.
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 45130
is hbexgVD 45479 without virtual deductions and was automatically derived
from hbexgVD 45479. (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 45480* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45143)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 45131 is ax6e2eqVD 45480 without virtual
deductions and was automatically derived from ax6e2eqVD 45480.
(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 45481* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45143)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2nd 45132 is ax6e2ndVD 45481 without virtual
deductions and was automatically derived from ax6e2ndVD 45481.
(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 45482* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45143)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 45131 is ax6e2ndeqVD 45482 without virtual
deductions and was automatically derived from ax6e2ndeqVD 45482.
(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 45483* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45143)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2sb5nd 45134 is 2sb5ndVD 45483 without virtual
deductions and was automatically derived from 2sb5ndVD 45483.
(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 45484* |
The following User's Proof is a Virtual Deduction proof (see wvd1 45143)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2uasbanh 45135 is 2uasbanhVD 45484 without
virtual deductions and was automatically derived from 2uasbanhVD 45484.
(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 45485 |
The following User's Proof is a Virtual Deduction proof (see wvd1 45143)
completed automatically by a Metamath tools program invoking mmj2 and the
Metamath Proof Assistant. e2ebind 45137 is e2ebindVD 45485 without virtual
deductions and was automatically derived from e2ebindVD 45485.
| 1:: | ⊢ (𝜑 ↔ 𝜑)
| | 2:1: | ⊢ (∀𝑦𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
| | 3:2: | ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑
))
| | 4:: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ ∀𝑦𝑦 = 𝑥 )
| | 5:3,4: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦𝜑 ↔ ∃𝑥
𝜑) )
| | 6:: | ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑦∀𝑦𝑦 = 𝑥)
| | 7:5,6: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ ∀𝑦(∃𝑦𝜑 ↔
∃𝑥𝜑) )
| | 8:7: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦∃𝑦𝜑 ↔
∃𝑦∃𝑥𝜑) )
| | 9:: | ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
| | 10:8,9: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦∃𝑦𝜑 ↔
∃𝑥∃𝑦𝜑) )
| | 11:: | ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
| | 12:11: | ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑)
| | 13:10,12: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑥∃𝑦𝜑 ↔
∃𝑦𝜑) )
| | 14:13: | ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃
𝑦𝜑))
| | 15:: | ⊢ (∀𝑦𝑦 = 𝑥 ↔ ∀𝑥𝑥 = 𝑦)
| | qed:14,15: | ⊢ (∀𝑥𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃
𝑦𝜑))
|
(Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
| |
| 21.41.8 Virtual Deduction transcriptions of
textbook proofs
|
| |
| Theorem | sb5ALTVD 45486* |
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 45099 is sb5ALTVD 45486 without virtual deductions and
was automatically derived from sb5ALTVD 45486.
| 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 45487 |
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 45102 is vk15.4jVD 45487
without virtual deductions and was automatically derived
from vk15.4jVD 45487. 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 45488 |
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 45103 is notnotrALTVD 45488
without virtual deductions and was automatically derived
from notnotrALTVD 45488. 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 45489 |
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 45104 is con3ALTVD 45489 without
virtual deductions and was automatically derived from con3ALTVD 45489.
Step i of the User's Proof corresponds to step i of the Fitch-style proof.
| 1:: | ⊢ ( (𝜑 → 𝜓) ▶ (𝜑 → 𝜓) )
| | 2:: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 )
| | 3:: | ⊢ (¬ ¬ 𝜑 → 𝜑)
| | 4:2: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜑 )
| | 5:1,4: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜓 )
| | 6:: | ⊢ (𝜓 → ¬ ¬ 𝜓)
| | 7:6,5: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜓 )
| | 8:7: | ⊢ ( (𝜑 → 𝜓) ▶ (¬ ¬ 𝜑 → ¬ ¬ 𝜓
) )
| | 9:: | ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 →
¬ 𝜑))
| | 10:8: | ⊢ ( (𝜑 → 𝜓) ▶ (¬ 𝜓 → ¬ 𝜑) )
| | qed:10: | ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) |
| |
| 21.41.9 Theorems proved using conjunction-form
Virtual Deduction
|
| |
| Theorem | elpwgdedVD 45490 |
Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived
from elpwg 4561. In form of VD deduction with 𝜑 and 𝜓 as
variable virtual hypothesis collections based on Mario Carneiro's
metavariable concept. elpwgded 45138 is elpwgdedVD 45490 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 45491 |
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 45491, using conventional notation, was
translated from virtual deduction form, sspwimpVD 45492, 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 45492 |
The following User's Proof is a Virtual Deduction proof (see wvd1 45143)
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 45491 is sspwimpVD 45492 without virtual deductions and was derived
from sspwimpVD 45492. (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 45493 |
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 45493, using
conventional notation, was translated from its virtual deduction form,
sspwimpcfVD 45494, 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 45494 |
The following User's Proof is a Virtual Deduction proof (see wvd1 45143)
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 45493 is sspwimpcfVD 45494 without virtual deductions and was derived
from sspwimpcfVD 45494.
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 45495 |
The successor of a transitive class is transitive. suctrALTcf 45495, using
conventional notation, was translated from virtual deduction form,
suctrALTcfVD 45496, 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 45496 |
The following User's Proof is a Virtual Deduction proof (see wvd1 45143)
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 45495
is suctrALTcfVD 45496 without virtual deductions and was derived
automatically from suctrALTcfVD 45496. The version of
completeusersproof.cmd used is capable of only generating
conjunction-form unification theorems, not unification deductions.
(Contributed by Alan Sare, 13-Jun-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( Tr 𝐴 ▶ Tr 𝐴 )
| | 2:: | ⊢ ( ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) )
| | 3:2: | ⊢ ( ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ 𝑧 ∈ 𝑦 )
| | 4:: | ⊢ ( ...................................
....... 𝑦 ∈ 𝐴 ▶ 𝑦 ∈ 𝐴 )
| | 5:1,3,4: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
, 𝑦 ∈ 𝐴 ) ▶ 𝑧 ∈ 𝐴 )
| | 6:: | ⊢ 𝐴 ⊆ suc 𝐴
| | 7:5,6: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
, 𝑦 ∈ 𝐴 ) ▶ 𝑧 ∈ suc 𝐴 )
| | 8:7: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
) ▶ (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) )
| | 9:: | ⊢ ( ...................................
...... 𝑦 = 𝐴 ▶ 𝑦 = 𝐴 )
| | 10:3,9: | ⊢ ( ........ ( (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴), 𝑦 = 𝐴 ) ▶ 𝑧 ∈ 𝐴 )
| | 11:10,6: | ⊢ ( ........ ( (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴), 𝑦 = 𝐴 ) ▶ 𝑧 ∈ suc 𝐴 )
| | 12:11: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) )
| | 13:2: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ 𝑦 ∈ suc 𝐴 )
| | 14:13: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) )
| | 15:8,12,14: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
) ▶ 𝑧 ∈ suc 𝐴 )
| | 16:15: | ⊢ ( Tr 𝐴 ▶ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) → 𝑧 ∈ suc 𝐴) )
| | 17:16: | ⊢ ( Tr 𝐴 ▶ ∀𝑧∀𝑦((𝑧 ∈
𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) )
| | 18:17: | ⊢ ( Tr 𝐴 ▶ Tr suc 𝐴 )
| | qed:18: | ⊢ (Tr 𝐴 → Tr suc 𝐴)
|
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| 21.41.10 Theorems with a VD proof in
conventional notation derived from a VD proof
|
| |
| Theorem | suctrALT3 45497 |
The successor of a transitive class is transitive. suctrALT3 45497 is the
completed proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/suctralt3vd.html 45497.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 45143 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 45140). 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 5214) .
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | sspwimpALT 45498 |
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 45498 is the completed
proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html 45498.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 45143 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 45138). 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 4565). (Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | unisnALT 45499 |
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 45499 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 45499. 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 45499, the
label list is a list of all 1-hypothesis propositional calculus
deductions in set.mm. stepproverp.c is the same as stepprover.c except
that it intermittently pauses during execution, allowing the User to
observe the changes to a text file caused by the execution of particular
statements of the program. (Contributed by Alan Sare, 19-Aug-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ∪
{𝐴} = 𝐴 |
| |
| 21.41.11 Theorems with a proof in conventional
notation derived from a VD proof
Theorems with a proof in conventional notation automatically derived by
completeusersproof.c from a Virtual Deduction User's Proof.
|
| |
| Theorem | notnotrALT2 45500 |
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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