Type | Label | Description |
Statement |
|
Theorem | suctrALT2 43901 |
Virtual deduction proof of suctr 6451. The sucessor of a transitive class
is transitive. This proof was generated automatically from the virtual
deduction proof suctrALT2VD 43900 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 43902* |
Virtual deduction proof of elex2 2811. (Contributed by Alan Sare,
25-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
|
Theorem | elex22VD 43903* |
Virtual deduction proof of elex22 3496. (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 43904* |
Virtual deduction proof of eqsbc2 3847. (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 43905* |
Virtual deduction proof of zfregs2 9731. (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 43906 |
Virtual deduction proof of tpid3g 4777. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
|
Theorem | en3lplem1VD 43907* |
Virtual deduction proof of en3lplem1 9610. (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 43908* |
Virtual deduction proof of en3lplem2 9611. (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 43909 |
Virtual deduction proof of en3lp 9612. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) |
|
21.39.7 Theorems proved using Virtual Deduction
with mmj2 assistance
|
|
Theorem | simplbi2VD 43910 |
Virtual deduction proof of simplbi2 500. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
h1:: | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
| 3:1,?: e0a 43836 | ⊢ ((𝜓 ∧ 𝜒) → 𝜑)
| qed:3,?: e0a 43836 | ⊢ (𝜓 → (𝜒 → 𝜑))
|
The proof of simplbi2 500 was automatically derived from it.
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 → (𝜒 → 𝜑)) |
|
Theorem | 3ornot23VD 43911 |
Virtual deduction proof of 3ornot23 43573. 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 43691 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜑 )
| 4:1,?: e1a 43691 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜓 )
| 5:3,4,?: e11 43752 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ (𝜑
∨ 𝜓) )
| 6:2,?: e2 43695 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ (𝜒 ∨ (𝜑 ∨ 𝜓)) )
| 7:5,6,?: e12 43788 | ⊢ ( (¬ 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 43912 |
Virtual deduction proof of orbi1r 43574. 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 43695 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜑 ∨ 𝜒) )
| 4:1,3,?: e12 43788 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜓 ∨ 𝜒) )
| 5:4,?: e2 43695 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜒 ∨ 𝜓) )
| 6:5: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜑)
→ (𝜒 ∨ 𝜓)) )
| 7:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜓) )
| 8:7,?: e2 43695 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜓 ∨ 𝜒) )
| 9:1,8,?: e12 43788 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜑 ∨ 𝜒) )
| 10:9,?: e2 43695 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜑) )
| 11:10: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜓)
→ (𝜒 ∨ 𝜑)) )
| 12:6,11,?: e11 43752 | ⊢ ( (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 43913 |
Virtual deduction proof of bitr3 351. 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 43691 | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜓
↔ 𝜑) )
| 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜑 ↔ 𝜒) )
| 4:3,?: e2 43695 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜒 ↔ 𝜑) )
| 5:2,4,?: e12 43788 | ⊢ ( (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 43914 |
Virtual deduction proof of 3orbi123 43575. 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 43691 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜑 ↔ 𝜓) )
| 3:1,?: e1a 43691 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜒 ↔ 𝜃) )
| 4:1,?: e1a 43691 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜏 ↔ 𝜂) )
| 5:2,3,?: e11 43752 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) )
| 6:5,4,?: e11 43752 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| 7:?: | ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ (𝜑
∨ 𝜒 ∨ 𝜏))
| 8:6,7,?: e10 43758 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| 9:?: | ⊢ (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔
(𝜓 ∨ 𝜃 ∨ 𝜂))
| 10:8,9,?: e10 43758 | ⊢ ( ((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 43915 |
Virtual deduction proof of the analogue of sbcor 3831 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 43691 | ⊢ ( 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 43758 | ⊢ ( 𝐴 ∈ 𝐵 ▶ [𝐴 / 𝑥](((𝜑
∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) )
| 4:1,33,?: e11 43752 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝜑
∨ 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒)) )
| 5:2,4,?: e11 43752 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒)) )
| 6:1,?: e1a 43691 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) )
| 7:6,?: e1a 43691 | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥](𝜑
∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| 8:5,7,?: e11 43752 | ⊢ ( 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 43758 | ⊢ ( 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 43916* |
Virtual deduction proof of alrim3con13v 43597. 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 43695 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜓 )
| 4:2,?: e2 43695 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜑 )
| 5:2,?: e2 43695 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜒 )
| 6:1,4,?: e12 43788 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜑 )
| 7:3,?: e2 43695 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜓 )
| 8:5,?: e2 43695 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜒 )
| 9:7,6,8,?: e222 43700 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒) )
| 10:9,?: e2 43695 | ⊢ ( (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 43917 |
Virtual deduction proof of exbir 43542. 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 43788 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓) ▶ (𝜒 ↔ 𝜃) )
| 6:3,5,?: e32 43822 | ⊢ ( ((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 43691 | ⊢ ( ((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 43918 |
Virtual deduction proof of exbiri 808. 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 43758 | ⊢ ( 𝜑 ▶ (𝜓 → (𝜒 ↔ 𝜃)) )
| 6:3,5,?: e21 43794 | ⊢ ( 𝜑 , 𝜓 ▶ (𝜒 ↔ 𝜃) )
| 7:4,6,?: e32 43822 | ⊢ ( 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 43919* |
Virtual deduction proof of rspsbc2 43598. 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 43812 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑 )
| 5:1,4,?: e13 43812 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑 )
| 6:2,5,?: e23 43819 | ⊢ ( 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 43920 |
Virtual deduction proof of 3impexp 1357. 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 43758 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| 4:3,?: e1a 43691 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| 5:4,?: e1a 43691 | ⊢ ( ((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 43691 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| 9:8,?: e1a 43691 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| 10:2,9,?: e01 43755 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) )
| 11:10: | ⊢ ((𝜑 → (𝜓 → (𝜒
→ 𝜃))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
| qed:6,11,?: e00 43832 | ⊢ (((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 43921 |
Virtual deduction proof of 3impexpbicom 43543. 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 43758 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃)) )
| 4:3,?: e1a 43691 | ⊢ ( ((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 43691 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏
↔ 𝜃)) )
| 8:7,2,?: e10 43758 | ⊢ ( (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 43832 | ⊢ (((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 43922 |
Virtual deduction proof of 3impexpbicomi 43544. 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 43836 | ⊢ (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 43923* |
Virtual deduction proof of sbcoreleleq 43599. 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 43691 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 ∈
𝑦 ↔ 𝑥 ∈ 𝐴) )
| 3:1,?: e1a 43691 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑦 ∈
𝑥 ↔ 𝐴 ∈ 𝑥) )
| 4:1,?: e1a 43691 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 =
𝑦 ↔ 𝑥 = 𝐴) )
| 5:2,3,4,?: e111 43738 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ((𝑥 ∈ 𝐴
∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥
∨ [𝐴 / 𝑦]𝑥 = 𝑦)) )
| 6:1,?: e1a 43691 | ⊢ ( 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 43752 | ⊢ ( 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 43924* |
Virtual deduction proof of nfra2 3371. 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 43832 | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 →
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| 4:2: | ⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| 5:4,?: e0a 43836 | ⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| qed:3,5,?: e00 43832 | ⊢ (∀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 43925* |
Virtual deduction proof of tratrb 43600. 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 43691 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| 3:1,?: e1a 43691 | ⊢ ( (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 43691 | ⊢ ( (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 43695 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝑦 )
| 7:5,?: e2 43695 | ⊢ ( (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 43720 | ⊢ ( (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 43717 | ⊢ ( (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 43699 | ⊢ ( (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 43791 | ⊢ ( (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 43819 | ⊢ ( (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 43819 | ⊢ ( (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 43791 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝑥 = 𝐵 )
| 21:3,?: e1a 43691 | ⊢ ( (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 43720 | ⊢ ( (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 43695 | ⊢ ( (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 43788 | ⊢ ( (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 43700 | ⊢ ( (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 43836 | ⊢ ((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 43836 | ⊢ ((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 43691 | ⊢ ( (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 43926 |
Virtual deduction proof of al2im 1815. 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 43691 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)) )
| 3:: | ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓
→ ∀𝑥𝜒))
| 4:2,3,?: e10 43758 | ⊢ ( ∀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 43927 |
Virtual deduction proof of syl5imp 43576. 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 43691 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜓
→ (𝜑 → 𝜒)) )
| 3:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → 𝜓) )
| 4:3,2,?: e21 43794 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → (𝜑 → 𝜒)) )
| 5:4,?: e2 43695 | ⊢ ( (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 43928 |
Virtual deduction proof of idiALT 43541. 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 43836 | ⊢ 𝜑
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ 𝜑 ⇒ ⊢ 𝜑 |
|
Theorem | ancomstVD 43929 |
Closed form of ancoms 458. The following user's proof is completed by
invoking mmj2's unify command and using mmj2's StepSelector to pick all
remaining steps of the Metamath proof.
1:: | ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
| qed:1,?: e0a 43836 | ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓
∧ 𝜑) → 𝜒))
|
The proof of ancomst 464 is derived automatically from it.
(Contributed by
Alan Sare, 25-Dec-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒)) |
|
Theorem | ssralv2VD 43930* |
Quantification restricted to a subclass for two quantifiers. ssralv 4051
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 43595 is ssralv2VD 43930 without
virtual deductions and was automatically derived from ssralv2VD 43930.
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 43931 |
An element of an ordinal class is ordinal. Proposition 7.6 of
[TakeutiZaring] p. 36. This is an alternate proof of ordelord 6387 using
the Axiom of Regularity indirectly through dford2 9618. 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 43601 is ordelordALTVD 43931
without virtual deductions and was automatically derived from
ordelordALTVD 43931 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 43932 |
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 4155 is equncomVD 43932 without
virtual deductions and was automatically derived from equncomVD 43932.
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 43933 |
Inference form of equncom 4155. 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 4156 is equncomiVD 43933 without
virtual deductions and was automatically derived from equncomiVD 43933.
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 43934 |
A set belongs to its successor. Alternate proof of sucid 6447.
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 43935 is sucidALTVD 43934
without virtual deductions and was automatically derived from
sucidALTVD 43934. 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 6371, 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 9618.
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 43935 |
A set belongs to its successor. This proof was automatically derived
from sucidALTVD 43934 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 43936 |
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 6447 is sucidVD 43936 without virtual deductions and was automatically
derived from sucidVD 43936.
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 43937 |
Implication form of imbi12i 349. 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 345 is imbi12VD 43937 without virtual
deductions and was automatically derived from imbi12VD 43937.
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 43938 |
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 43584
is imbi13VD 43938 without virtual deductions and was automatically derived
from imbi13VD 43938.
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 43939 |
Distribution of class substitution over a left-nested implication.
Similar to sbcimg 3829.
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 43602 is sbcim2gVD 43939 without virtual deductions and was automatically
derived from sbcim2gVD 43939.
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 43940 |
Implication form of sbcbii 3838.
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 43603 is sbcbiVD 43940 without virtual deductions and was automatically
derived from sbcbiVD 43940.
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 43941* |
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 43604 is trsbcVD 43941 without virtual deductions and was automatically
derived from trsbcVD 43941.
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 43942* |
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 43605 is truniALTVD 43942 without virtual deductions and was
automatically derived from truniALTVD 43942.
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 43943 |
Non-virtual deduction form of e33 43798.
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 43585 is ee33VD 43943 without virtual deductions and was automatically
derived from ee33VD 43943.
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 43944* |
The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 43945.
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 43945 is trintALTVD 43944 without virtual deductions and was
automatically derived from trintALTVD 43944.
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 43945* |
The intersection of a class of transitive sets is transitive. Exercise
5(b) of [Enderton] p. 73. trintALT 43945 is an alternate proof of trint 5284.
trintALT 43945 is trintALTVD 43944 without virtual deductions and was
automatically derived from trintALTVD 43944 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 43946 |
The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual
deduction proof of undif3 4291.
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 4291 is undif3VD 43946 without virtual deductions and was automatically
derived from undif3VD 43946.
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 43947 |
Virtual deduction proof of sbcssg 4524.
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 4524 is sbcssgVD 43947 without virtual deductions and was automatically
derived from sbcssgVD 43947.
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 43948 |
Virtual deduction proof of csbin 4440.
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 4440 is csbingVD 43948 without virtual deductions and was
automatically derived from csbingVD 43948.
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 43949* |
Virtual deduction proof of onfrALTlem5 43606.
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 43606 is onfrALTlem5VD 43949 without virtual deductions and was
automatically derived from onfrALTlem5VD 43949.
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 43950* |
Virtual deduction proof of onfrALTlem4 43607.
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 43607 is onfrALTlem4VD 43950 without virtual deductions and was
automatically derived from onfrALTlem4VD 43950.
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 43951* |
Virtual deduction proof of onfrALTlem3 43608.
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 43608 is onfrALTlem3VD 43951 without virtual deductions and was
automatically derived from onfrALTlem3VD 43951.
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 43952 |
Virtual deduction proof of simplbi2comt 501.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
simplbi2comt 501 is simplbi2comtVD 43952 without virtual deductions and was
automatically derived from simplbi2comtVD 43952.
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 43953* |
Virtual deduction proof of onfrALTlem2 43610.
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 43610 is onfrALTlem2VD 43953 without virtual deductions and was
automatically derived from onfrALTlem2VD 43953.
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 43954* |
Virtual deduction proof of onfrALTlem1 43612.
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 43612 is onfrALTlem1VD 43954 without virtual deductions and was
automatically derived from onfrALTlem1VD 43954.
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 43955 |
Virtual deduction proof of onfrALT 43613.
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 43613 is onfrALTVD 43955 without virtual deductions and was
automatically derived from onfrALTVD 43955.
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 43956 |
Virtual deduction proof of csbeq2 3899.
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 3899 is csbeq2gVD 43956 without virtual deductions and was
automatically derived from csbeq2gVD 43956.
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 43957 |
Virtual deduction proof of csbsng 4713.
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 4713 is csbsngVD 43957 without virtual deductions and was automatically
derived from csbsngVD 43957.
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 43958 |
Virtual deduction proof of csbxp 5776.
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 5776 is csbxpgVD 43958 without virtual deductions and was
automatically derived from csbxpgVD 43958.
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 43959 |
Virtual deduction proof of csbres 5985.
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 5985 is csbresgVD 43959 without virtual deductions and was
automatically derived from csbresgVD 43959.
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 43960 |
Virtual deduction proof of csbrn 6203.
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 6203 is csbrngVD 43960 without virtual deductions and was
automatically derived from csbrngVD 43960.
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 43961 |
Virtual deduction proof of csbima12 6079.
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 6079 is csbima12gALTVD 43961 without virtual deductions and was
automatically derived from csbima12gALTVD 43961.
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 43962 |
Virtual deduction proof of csbuni 4941.
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 4941 is csbunigVD 43962 without virtual deductions and was
automatically derived from csbunigVD 43962.
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 43963 |
Virtual deduction proof of csbfv12 6940.
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 6940 is csbfv12gALTVD 43963 without virtual deductions and was
automatically derived from csbfv12gALTVD 43963.
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 43964 |
Virtual deduction proof of con5 43586.
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 43586 is con5VD 43964 without virtual deductions and was automatically
derived from con5VD 43964.
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 43965 |
Virtual deduction proof of relopab 5825.
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 5825 is relopabVD 43965 without virtual deductions and was
automatically derived from relopabVD 43965.
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 43966 |
Virtual deduction proof of 19.41rg 43614.
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 43614
is 19.41rgVD 43966 without virtual deductions and was automatically derived
from 19.41rgVD 43966. (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 43967 |
Virtual deduction proof of 2pm13.193 43616.
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 43616 is 2pm13.193VD 43967 without virtual deductions and was
automatically derived from 2pm13.193VD 43967. (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 43968 |
Virtual deduction proof of hbimpg 43618.
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 43618
is hbimpgVD 43968 without virtual deductions and was automatically derived
from hbimpgVD 43968. (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 43969 |
Virtual deduction proof of hbalg 43619.
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 43619
is hbalgVD 43969 without virtual deductions and was automatically derived
from hbalgVD 43969. (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 43970 |
Virtual deduction proof of hbexg 43620.
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 43620
is hbexgVD 43970 without virtual deductions and was automatically derived
from hbexgVD 43970. (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 43971* |
The following User's Proof is a Virtual Deduction proof (see wvd1 43633)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 43621 is ax6e2eqVD 43971 without virtual
deductions and was automatically derived from ax6e2eqVD 43971.
(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 43972* |
The following User's Proof is a Virtual Deduction proof (see wvd1 43633)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2nd 43622 is ax6e2ndVD 43972 without virtual
deductions and was automatically derived from ax6e2ndVD 43972.
(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 43973* |
The following User's Proof is a Virtual Deduction proof (see wvd1 43633)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 43621 is ax6e2ndeqVD 43973 without virtual
deductions and was automatically derived from ax6e2ndeqVD 43973.
(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 43974* |
The following User's Proof is a Virtual Deduction proof (see wvd1 43633)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2sb5nd 43624 is 2sb5ndVD 43974 without virtual
deductions and was automatically derived from 2sb5ndVD 43974.
(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 43975* |
The following User's Proof is a Virtual Deduction proof (see wvd1 43633)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2uasbanh 43625 is 2uasbanhVD 43975 without
virtual deductions and was automatically derived from 2uasbanhVD 43975.
(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 43976 |
The following User's Proof is a Virtual Deduction proof (see wvd1 43633)
completed automatically by a Metamath tools program invoking mmj2 and the
Metamath Proof Assistant. e2ebind 43627 is e2ebindVD 43976 without virtual
deductions and was automatically derived from e2ebindVD 43976.
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)) |
|
21.39.8 Virtual Deduction transcriptions of
textbook proofs
|
|
Theorem | sb5ALTVD 43977* |
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 2266, 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 43589 is sb5ALTVD 43977 without virtual deductions and
was automatically derived from sb5ALTVD 43977.
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 43978 |
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 43592 is vk15.4jVD 43978
without virtual deductions and was automatically derived
from vk15.4jVD 43978. 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 43979 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Theorem 5 of
Section 14 of [Margaris] p. 59 (which is notnotr 130). The same proof
may also be interpreted as a Virtual Deduction Hilbert-style
axiomatic proof. It was completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. notnotrALT 43593 is notnotrALTVD 43979
without virtual deductions and was automatically derived
from notnotrALTVD 43979. 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 43980 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Theorem 7 of
Section 14 of [Margaris] p. 60 (which is con3 153). The same proof may
also be interpreted to be a Virtual Deduction Hilbert-style axiomatic
proof. It was completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. con3ALT2 43594 is con3ALTVD 43980 without
virtual deductions and was automatically derived from con3ALTVD 43980.
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)) |
|
21.39.9 Theorems proved using conjunction-form
Virtual Deduction
|
|
Theorem | elpwgdedVD 43981 |
Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived
from elpwg 4606. In form of VD deduction with 𝜑 and 𝜓 as
variable virtual hypothesis collections based on Mario Carneiro's
metavariable concept. elpwgded 43628 is elpwgdedVD 43981 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 43982 |
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 5450. The proof sspwimp 43982, using conventional notation, was
translated from virtual deduction form, sspwimpVD 43983, 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 43983 |
The following User's Proof is a Virtual Deduction proof (see wvd1 43633)
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 43982 is sspwimpVD 43983 without virtual deductions and was derived
from sspwimpVD 43983. (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 43984 |
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 43984, using
conventional notation, was translated from its virtual deduction form,
sspwimpcfVD 43985, 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 43985 |
The following User's Proof is a Virtual Deduction proof (see wvd1 43633)
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 43984 is sspwimpcfVD 43985 without virtual deductions and was derived
from sspwimpcfVD 43985.
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 43986 |
The sucessor of a transitive class is transitive. suctrALTcf 43986, using
conventional notation, was translated from virtual deduction form,
suctrALTcfVD 43987, 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 43987 |
The following User's Proof is a Virtual Deduction proof (see wvd1 43633)
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 43986
is suctrALTcfVD 43987 without virtual deductions and was derived
automatically from suctrALTcfVD 43987. 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) |
|
21.39.10 Theorems with a VD proof in
conventional notation derived from a VD proof
|
|
Theorem | suctrALT3 43988 |
The successor of a transitive class is transitive. suctrALT3 43988 is the
completed proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/suctralt3vd.html 43988.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 43633 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 43630). 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 5268) .
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (Tr 𝐴 → Tr suc 𝐴) |
|
Theorem | sspwimpALT 43989 |
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 43989 is the completed
proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html 43989.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 43633 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 43628). 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 4610). (Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
|
Theorem | unisnALT 43990 |
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 43990 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 43990. 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 43990, 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 |
|
21.39.11 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 43991 |
Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102.
Proof derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. (Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (¬ ¬
𝜑 → 𝜑) |
|
Theorem | sspwimpALT2 43992 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. Proof derived by
completeusersproof.c from User's Proof in VirtualDeductionProofs.txt.
The User's Proof in html format is displayed in
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html.
(Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
|
Theorem | e2ebindALT 43993 |
Absorption of an existential quantifier of a double existential quantifier
of non-distinct variables. The proof is derived by completeusersproof.c
from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html
format is displayed in e2ebindVD 43976. (Contributed by Alan Sare,
11-Sep-2016.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
|
Theorem | ax6e2ndALT 43994* |
If at least two sets exist (dtru 5437), then the same is true expressed
in an alternate form similar to the form of ax6e 2381.
The proof is
derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. The User's Proof in html format is
displayed in ax6e2ndVD 43972. (Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
|
Theorem | ax6e2ndeqALT 43995* |
"At least two sets exist" expressed in the form of dtru 5437
is logically
equivalent to the same expressed in a form similar to ax6e 2381
if dtru 5437
is false implies 𝑢 = 𝑣. Proof derived by
completeusersproof.c from
User's Proof in VirtualDeductionProofs.txt. The User's Proof in html
format is displayed in ax6e2ndeqVD 43973. (Contributed by Alan Sare,
11-Sep-2016.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
|
Theorem | 2sb5ndALT 43996* |
Equivalence for double substitution 2sb5 2270 without distinct 𝑥,
𝑦 requirement. 2sb5nd 43624 is derived from 2sb5ndVD 43974. The proof is
derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. The User's Proof in html format is
displayed in 2sb5ndVD 43974. (Contributed by Alan Sare, 19-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) |
|
Theorem | chordthmALT 43997* |
The intersecting chords theorem. If points A, B, C, and D lie on a
circle (with center Q, say), and the point P is on the interior of the
segments AB and CD, then the two products of lengths PA · PB and
PC · PD are equal. The Euclidean plane is identified with the
complex plane, and the fact that P is on AB and on CD is expressed by
the hypothesis that the angles APB and CPD are equal to π. The
result is proven by using chordthmlem5 26574 twice to show that PA
· PB and PC · PD both equal BQ
2
−
PQ
2
. This is similar to the proof of the
theorem given in Euclid's Elements, where it is Proposition
III.35.
Proven by David Moews on 28-Feb-2017 as chordthm 26575.
https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 26575 is
a Virtual
Deduction User's Proof transcription of chordthm 26575. That VD User's
Proof was input into completeusersproof, automatically generating this
chordthmALT 43997 Metamath proof. (Contributed by Alan Sare,
19-Sep-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0})
↦ (ℑ‘(log‘(𝑦 / 𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑃 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝑃)
& ⊢ (𝜑 → 𝐵 ≠ 𝑃)
& ⊢ (𝜑 → 𝐶 ≠ 𝑃)
& ⊢ (𝜑 → 𝐷 ≠ 𝑃)
& ⊢ (𝜑 → ((𝐴 − 𝑃)𝐹(𝐵 − 𝑃)) = π) & ⊢ (𝜑 → ((𝐶 − 𝑃)𝐹(𝐷 − 𝑃)) = π) & ⊢ (𝜑 → 𝑄 ∈ ℂ) & ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄))) & ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐶 − 𝑄))) & ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐷 − 𝑄))) ⇒ ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = ((abs‘(𝑃 − 𝐶)) · (abs‘(𝑃 − 𝐷)))) |
|
Theorem | isosctrlem1ALT 43998 |
Lemma for isosctr 26559. This proof was automatically derived by
completeusersproof from its Virtual Deduction proof counterpart
https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 26559.
As it is verified by the Metamath program, isosctrlem1ALT 43998 verifies
https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 43998.
(Contributed by Alan Sare, 22-Apr-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ ((𝐴 ∈ ℂ ∧
(abs‘𝐴) = 1 ∧
¬ 1 = 𝐴) →
(ℑ‘(log‘(1 − 𝐴))) ≠ π) |
|
Theorem | iunconnlem2 43999* |
The indexed union of connected overlapping subspaces sharing a common
point is connected. This proof was automatically derived by
completeusersproof from its Virtual Deduction proof counterpart
https://us.metamath.org/other/completeusersproof/iunconlem2vd.html.
As it is verified by the Metamath program, iunconnlem2 43999 verifies
https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 43999.
(Contributed by Alan Sare, 22-Apr-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝜓 ↔ ((((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ (𝑢 ∩ ∪
𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑣 ∩ ∪
𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑢 ∩ 𝑣) ⊆ (𝑋 ∖ ∪ 𝑘 ∈ 𝐴 𝐵)) ∧ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ (𝑢 ∪ 𝑣))) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)
⇒ ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn) |
|
Theorem | iunconnALT 44000* |
The indexed union of connected overlapping subspaces sharing a common
point is connected. This proof was automatically derived by
completeusersproof from its Virtual Deduction proof counterpart
https://us.metamath.org/other/completeusersproof/iunconaltvd.html.
As it is verified by the Metamath program, iunconnALT 44000 verifies
https://us.metamath.org/other/completeusersproof/iunconaltvd.html 44000.
(Contributed by Alan Sare, 22-Apr-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)
⇒ ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn) |