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Theorem List for Metamath Proof Explorer - 43601-43700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeqsbc2VD 43601* Virtual deduction proof of eqsbc2 3847. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
 
Theoremzfregs2VD 43602* Virtual deduction proof of zfregs2 9728. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≠ ∅ → ¬ ∀𝑥𝐴𝑦(𝑦𝐴𝑦𝑥))
 
Theoremtpid3gVD 43603 Virtual deduction proof of tpid3g 4777. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 
Theoremen3lplem1VD 43604* Virtual deduction proof of en3lplem1 9607. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
 
Theoremen3lplem2VD 43605* Virtual deduction proof of en3lplem2 9608. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
 
Theoremen3lpVD 43606 Virtual deduction proof of en3lp 9609. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
 
21.39.7  Theorems proved using Virtual Deduction with mmj2 assistance
 
Theoremsimplbi2VD 43607 Virtual deduction proof of simplbi2 502. 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 ↔ (𝜓𝜒))
3:1,?: e0a 43533 ((𝜓𝜒) → 𝜑)
qed:3,?: e0a 43533 (𝜓 → (𝜒𝜑))
The proof of simplbi2 502 was automatically derived from it. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
 
Theorem3ornot23VD 43608 Virtual deduction proof of 3ornot23 43270. 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)   ,   (𝜒𝜑 𝜓)   ▶   (𝜒𝜑𝜓)   )
3:1,?: e1a 43388 (   𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜑   )
4:1,?: e1a 43388 (   𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜓   )
5:3,4,?: e11 43449 (   𝜑 ∧ ¬ 𝜓)   ▶   ¬ (𝜑 𝜓)   )
6:2,?: e2 43392 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑 𝜓)   ▶   (𝜒 ∨ (𝜑𝜓))   )
7:5,6,?: e12 43485 (   𝜑 ∧ ¬ 𝜓)   ,   (𝜒𝜑 𝜓)   ▶   𝜒   )
8:7: (   𝜑 ∧ ¬ 𝜓)   ▶   ((𝜒 𝜑𝜓) → 𝜒)   )
qed:8: ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 𝜑𝜓) → 𝜒))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))
 
Theoremorbi1rVD 43609 Virtual deduction proof of orbi1r 43271. 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:: (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2:: (   (𝜑𝜓)   ,   (𝜒𝜑)    ▶   (𝜒𝜑)   )
3:2,?: e2 43392 (   (𝜑𝜓)   ,   (𝜒𝜑)    ▶   (𝜑𝜒)   )
4:1,3,?: e12 43485 (   (𝜑𝜓)   ,   (𝜒𝜑)    ▶   (𝜓𝜒)   )
5:4,?: e2 43392 (   (𝜑𝜓)   ,   (𝜒𝜑)    ▶   (𝜒𝜓)   )
6:5: (   (𝜑𝜓)   ▶   ((𝜒𝜑) → (𝜒𝜓))   )
7:: (   (𝜑𝜓)   ,   (𝜒𝜓)    ▶   (𝜒𝜓)   )
8:7,?: e2 43392 (   (𝜑𝜓)   ,   (𝜒𝜓)    ▶   (𝜓𝜒)   )
9:1,8,?: e12 43485 (   (𝜑𝜓)   ,   (𝜒𝜓)    ▶   (𝜑𝜒)   )
10:9,?: e2 43392 (   (𝜑𝜓)   ,   (𝜒𝜓)    ▶   (𝜒𝜑)   )
11:10: (   (𝜑𝜓)   ▶   ((𝜒𝜓) → (𝜒𝜑))   )
12:6,11,?: e11 43449 (   (𝜑𝜓)   ▶   ((𝜒 𝜑) ↔ (𝜒𝜓))   )
qed:12: ((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
 
Theorembitr3VD 43610 Virtual deduction proof of bitr3 353. 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:: (   (𝜑𝜓)   ▶   (𝜑 𝜓)   )
2:1,?: e1a 43388 (   (𝜑𝜓)   ▶   (𝜓 𝜑)   )
3:: (   (𝜑𝜓)   ,   (𝜑𝜒)    ▶   (𝜑𝜒)   )
4:3,?: e2 43392 (   (𝜑𝜓)   ,   (𝜑𝜒)    ▶   (𝜒𝜑)   )
5:2,4,?: e12 43485 (   (𝜑𝜓)   ,   (𝜑𝜒)    ▶   (𝜓𝜒)   )
6:5: (   (𝜑𝜓)   ▶   ((𝜑 𝜒) → (𝜓𝜒))   )
qed:6: ((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))
 
Theorem3orbi123VD 43611 Virtual deduction proof of 3orbi123 43272. 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:: (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   )
2:1,?: e1a 43388 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   (𝜑𝜓)   )
3:1,?: e1a 43388 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   (𝜒𝜃)   )
4:1,?: e1a 43388 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   (𝜏𝜂)   )
5:2,3,?: e11 43449 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   ((𝜑𝜒) ↔ (𝜓𝜃))   )
6:5,4,?: e11 43449 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   (((𝜑𝜒) ∨ 𝜏) ↔ ((𝜓𝜃) 𝜂))   )
7:?: (((𝜑𝜒) ∨ 𝜏) ↔ (𝜑 𝜒𝜏))
8:6,7,?: e10 43455 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   ((𝜑𝜒𝜏) ↔ ((𝜓𝜃) 𝜂))   )
9:?: (((𝜓𝜃) ∨ 𝜂) ↔ (𝜓𝜃𝜂))
10:8,9,?: e10 43455 (   ((𝜑𝜓) ∧ (𝜒 𝜃) ∧ (𝜏𝜂))   ▶   ((𝜑𝜒𝜏) ↔ (𝜓 𝜃𝜂))   )
qed:10: (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((𝜑𝜒𝜏) ↔ (𝜓𝜃 𝜂)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂)))
 
Theoremsbc3orgVD 43612 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:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:1,?: e1a 43388 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑 𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) [𝐴 / 𝑥]𝜒))   )
3:: (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 𝜓𝜒))
32:3: 𝑥(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))
33:1,32,?: e10 43455 (   𝐴𝐵   ▶   [𝐴 / 𝑥](((𝜑 𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))   )
4:1,33,?: e11 43449 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))   )
5:2,4,?: e11 43449 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
6:1,?: e1a 43388 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
7:6,?: e1a 43388 (   𝐴𝐵   ▶   (([𝐴 / 𝑥](𝜑 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) [𝐴 / 𝑥]𝜒))   )
8:5,7,?: e11 43449 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ (([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 43455 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒))   )
qed:10: (𝐴𝐵 → ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
 
Theorem19.21a3con13vVD 43613* Virtual deduction proof of alrim3con13v 43294. 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)   ,   (𝜓𝜑 𝜒)   ▶   (𝜓𝜑𝜒)   )
3:2,?: e2 43392 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝜓   )
4:2,?: e2 43392 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝜑   )
5:2,?: e2 43392 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝜒   )
6:1,4,?: e12 43485 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝑥𝜑   )
7:3,?: e2 43392 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝑥𝜓   )
8:5,?: e2 43392 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝑥𝜒   )
9:7,6,8,?: e222 43397 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒)   )
10:9,?: e2 43392 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝑥(𝜓𝜑𝜒)   )
11:10:in2 (   (𝜑 → ∀𝑥𝜑)   ▶   ((𝜓 𝜑𝜒) → ∀𝑥(𝜓𝜑𝜒))   )
qed:11:in1 ((𝜑 → ∀𝑥𝜑) → ((𝜓 𝜑𝜒) → ∀𝑥(𝜓𝜑𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ∀𝑥𝜑) → ((𝜓𝜑𝜒) → ∀𝑥(𝜓𝜑𝜒)))
 
TheoremexbirVD 43614 Virtual deduction proof of exbir 43239. 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:: (   ((𝜑𝜓) → (𝜒𝜃))    ▶   ((𝜑𝜓) → (𝜒𝜃))   )
2:: (   ((𝜑𝜓) → (𝜒𝜃))   ,    (𝜑𝜓)   ▶   (𝜑𝜓)   )
3:: (   ((𝜑𝜓) → (𝜒𝜃))   ,    (𝜑𝜓), 𝜃   ▶   𝜃   )
5:1,2,?: e12 43485 (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓)   ▶   (𝜒𝜃)   )
6:3,5,?: e32 43519 (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓), 𝜃   ▶   𝜒   )
7:6: (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓)   ▶   (𝜃𝜒)   )
8:7: (   ((𝜑𝜓) → (𝜒𝜃))    ▶   ((𝜑𝜓) → (𝜃𝜒))   )
9:8,?: e1a 43388 (   ((𝜑𝜓) → (𝜒 𝜃))   ▶   (𝜑 → (𝜓 → (𝜃𝜒)))   )
qed:9: (((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))
(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))
 
TheoremexbiriVD 43615 Virtual deduction proof of exbiri 810. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1:: ((𝜑𝜓) → (𝜒𝜃))
2:: (   𝜑   ▶   𝜑   )
3:: (   𝜑   ,   𝜓   ▶   𝜓   )
4:: (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   )
5:2,1,?: e10 43455 (   𝜑   ▶   (𝜓 → (𝜒𝜃))   )
6:3,5,?: e21 43491 (   𝜑   ,   𝜓   ▶   (𝜒𝜃)   )
7:4,6,?: e32 43519 (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
8:7: (   𝜑   ,   𝜓   ▶   (𝜃𝜒)   )
9:8: (   𝜑   ▶   (𝜓 → (𝜃𝜒))   )
qed:9: (𝜑 → (𝜓 → (𝜃𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theoremrspsbc2VD 43616* Virtual deduction proof of rspsbc2 43295. 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:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (   𝐴𝐵   ,   𝐶𝐷   ▶   𝐶𝐷   )
3:: (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥𝐵𝑦𝐷𝜑   )
4:1,3,?: e13 43509 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   [𝐴 / 𝑥]𝑦𝐷𝜑   )
5:1,4,?: e13 43509 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑦𝐷[𝐴 / 𝑥]𝜑   )
6:2,5,?: e23 43516 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   )
7:6: (   𝐴𝐵   ,   𝐶𝐷   ▶   (∀𝑥𝐵 𝑦𝐷𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   )
8:7: (   𝐴𝐵   ▶   (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   )
qed:8: (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
 
Theorem3impexpVD 43617 Virtual deduction proof of 3impexp 1359. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1:: (   ((𝜑𝜓𝜒) 𝜃)   ▶   ((𝜑𝜓𝜒) → 𝜃)   )
2:: ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
3:1,2,?: e10 43455 (   ((𝜑𝜓𝜒) 𝜃)   ▶   (((𝜑𝜓) ∧ 𝜒) → 𝜃)   )
4:3,?: e1a 43388 (   ((𝜑𝜓𝜒) 𝜃)   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
5:4,?: e1a 43388 (   ((𝜑𝜓𝜒) 𝜃)   ▶   (𝜑 → (𝜓 → (𝜒𝜃)))   )
6:5: (((𝜑𝜓𝜒) → 𝜃) → (𝜑 → (𝜓 → (𝜒𝜃))))
7:: (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   (𝜑 → (𝜓 → (𝜒𝜃)))   )
8:7,?: e1a 43388 (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
9:8,?: e1a 43388 (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   (((𝜑𝜓) ∧ 𝜒) → 𝜃)   )
10:2,9,?: e01 43452 (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   ((𝜑𝜓𝜒) → 𝜃)   )
11:10: ((𝜑 → (𝜓 → (𝜒 𝜃))) → ((𝜑𝜓𝜒) → 𝜃))
qed:6,11,?: e00 43529 (((𝜑𝜓𝜒) 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
 
Theorem3impexpbicomVD 43618 Virtual deduction proof of 3impexpbicom 43240. 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:: (   ((𝜑𝜓𝜒) → (𝜃𝜏))   ▶   ((𝜑𝜓𝜒) → (𝜃𝜏))   )
2:: ((𝜃𝜏) ↔ (𝜏 𝜃))
3:1,2,?: e10 43455 (   ((𝜑𝜓𝜒) → (𝜃𝜏))   ▶   ((𝜑𝜓𝜒) → (𝜏𝜃))   )
4:3,?: e1a 43388 (   ((𝜑𝜓𝜒) → (𝜃𝜏))   ▶   (𝜑 → (𝜓 → (𝜒 → (𝜏 𝜃))))   )
5:4: (((𝜑𝜓𝜒) → (𝜃𝜏)) → (𝜑 → (𝜓 → (𝜒 → (𝜏 𝜃)))))
6:: (   (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))   ▶   (𝜑 → (𝜓 → (𝜒 → (𝜏 𝜃))))   )
7:6,?: e1a 43388 (   (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))   ▶   ((𝜑𝜓𝜒) → (𝜏 𝜃))   )
8:7,2,?: e10 43455 (   (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))   ▶   ((𝜑𝜓𝜒) → (𝜃 𝜏))   )
9:8: ((𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))) → ((𝜑𝜓𝜒) → (𝜃 𝜏)))
qed:5,9,?: e00 43529 (((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 𝜃)))))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
 
Theorem3impexpbicomiVD 43619 Virtual deduction proof of 3impexpbicomi 43241. 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:: ((𝜑𝜓𝜒) → (𝜃 𝜏))
qed:1,?: e0a 43533 (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜒) → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))
 
TheoremsbcoreleleqVD 43620* Virtual deduction proof of sbcoreleleq 43296. 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:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:1,?: e1a 43388 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 𝑦𝑥𝐴)   )
3:1,?: e1a 43388 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑦 𝑥𝐴𝑥)   )
4:1,?: e1a 43388 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)   )
5:2,3,4,?: e111 43435 (   𝐴𝐵   ▶   ((𝑥𝐴 𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥 [𝐴 / 𝑦]𝑥 = 𝑦))   )
6:1,?: e1a 43388 (   𝐴𝐵    ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥 𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
7:5,6: e11 43449 (   𝐴𝐵   ▶   ([𝐴 / 𝑦](𝑥 𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴))   )
qed:7: (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦 𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
 
Theoremhbra2VD 43621* Virtual deduction proof of nfra2 3373. 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:: (∀𝑦𝐵𝑥𝐴𝜑 𝑦𝑦𝐵𝑥𝐴𝜑)
2:: (∀𝑥𝐴𝑦𝐵𝜑 𝑦𝐵𝑥𝐴𝜑)
3:1,2,?: e00 43529 (∀𝑥𝐴𝑦𝐵𝜑 𝑦𝑦𝐵𝑥𝐴𝜑)
4:2: 𝑦(∀𝑥𝐴𝑦𝐵𝜑 𝑦𝐵𝑥𝐴𝜑)
5:4,?: e0a 43533 (∀𝑦𝑥𝐴𝑦𝐵𝜑 𝑦𝑦𝐵𝑥𝐴𝜑)
qed:3,5,?: e00 43529 (∀𝑥𝐴𝑦𝐵𝜑 𝑦𝑥𝐴𝑦𝐵𝜑)
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝑥𝐴𝑦𝐵 𝜑)
 
TheoremtratrbVD 43622* Virtual deduction proof of tratrb 43297. 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 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)    ▶   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) 𝐵𝐴)   )
2:1,?: e1a 43388 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐴   )
3:1,?: e1a 43388 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)    ▶   𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
4:1,?: e1a 43388 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝐵𝐴   )
5:: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥𝑦𝑦𝐵)   )
6:5,?: e2 43392 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑥𝑦   )
7:5,?: e2 43392 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑦𝐵   )
8:2,7,4,?: e121 43417 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑦𝐴   )
9:2,6,8,?: e122 43414 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑥𝐴   )
10:: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝐵𝑥   ▶   𝐵𝑥   )
11:6,7,10,?: e223 43396 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝐵𝑥   ▶   (𝑥𝑦𝑦𝐵𝐵𝑥)   )
12:11: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))   )
13:: ¬ (𝑥𝑦𝑦𝐵 𝐵𝑥)
14:12,13,?: e20 43488 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   ¬ 𝐵𝑥   )
15:: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
16:7,15,?: e23 43516 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝑥 = 𝐵   ▶   𝑦𝑥   )
17:6,16,?: e23 43516 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝑥 = 𝐵   ▶   (𝑥𝑦𝑦𝑥)   )
18:17: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥))   )
19:: ¬ (𝑥𝑦𝑦𝑥)
20:18,19,?: e20 43488 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   ¬ 𝑥 = 𝐵   )
21:3,?: e1a 43388 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦𝐴 𝑥𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
22:21,9,4,?: e121 43417 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥 𝑥 = 𝑦)   )
23:22,?: e2 43392 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
24:4,23,?: e12 43485 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥𝐵𝐵𝑥𝑥 = 𝐵)   )
25:14,20,24,?: e222 43397 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑥𝐵   )
26:25: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   ((𝑥𝑦 𝑦𝐵) → 𝑥𝐵)   )
27:: (∀𝑥𝐴𝑦𝐴(𝑥𝑦 𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑥𝐴𝑦𝐴(𝑥𝑦 𝑦𝑥𝑥 = 𝑦))
28:27,?: e0a 43533 ((Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥 𝑥 = 𝑦) ∧ 𝐵𝐴))
29:28,26: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)    ▶   𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
30:: (∀𝑥𝐴𝑦𝐴(𝑥𝑦 𝑦𝑥𝑥 = 𝑦) → ∀𝑥𝑥𝐴𝑦𝐴(𝑥𝑦 𝑦𝑥𝑥 = 𝑦))
31:30,?: e0a 43533 ((Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴))
32:31,29: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑥 𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
33:32,?: e1a 43388 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐵   )
qed:33: ((Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
 
Theoremal2imVD 43623 Virtual deduction proof of al2im 1817. 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 → (𝜓𝜒))    ▶   𝑥(𝜑 → (𝜓𝜒))   )
2:1,?: e1a 43388 (   𝑥(𝜑 → (𝜓𝜒))    ▶   (∀𝑥𝜑 → ∀𝑥(𝜓𝜒))   )
3:: (∀𝑥(𝜓𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒))
4:2,3,?: e10 43455 (   𝑥(𝜑 → (𝜓𝜒))    ▶   (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))   )
qed:4: (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
 
Theoremsyl5impVD 43624 Virtual deduction proof of syl5imp 43273. 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 → (𝜓𝜒))   ▶   (𝜑 → (𝜓𝜒))   )
2:1,?: e1a 43388 (   (𝜑 → (𝜓𝜒))   ▶   (𝜓 → (𝜑𝜒))   )
3:: (   (𝜑 → (𝜓𝜒))   ,   (𝜃 𝜓)   ▶   (𝜃𝜓)   )
4:3,2,?: e21 43491 (   (𝜑 → (𝜓𝜒))   ,   (𝜃 𝜓)   ▶   (𝜃 → (𝜑𝜒))   )
5:4,?: e2 43392 (   (𝜑 → (𝜓𝜒))   ,   (𝜃 𝜓)   ▶   (𝜑 → (𝜃𝜒))   )
6:5: (   (𝜑 → (𝜓𝜒))   ▶   ((𝜃 𝜓) → (𝜑 → (𝜃𝜒)))   )
qed:6: ((𝜑 → (𝜓𝜒)) → ((𝜃 𝜓) → (𝜑 → (𝜃𝜒))))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜓) → (𝜑 → (𝜃𝜒))))
 
TheoremidiVD 43625 Virtual deduction proof of idiALT 43238. 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 43533 𝜑
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑       𝜑
 
TheoremancomstVD 43626 Closed form of ancoms 460. 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:: ((𝜑𝜓) ↔ (𝜓𝜑))
qed:1,?: e0a 43533 (((𝜑𝜓) → 𝜒) ↔ ((𝜓 𝜑) → 𝜒))
The proof of ancomst 466 is derived automatically from it. (Contributed by Alan Sare, 25-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))
 
Theoremssralv2VD 43627* 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 43292 is ssralv2VD 43627 without virtual deductions and was automatically derived from ssralv2VD 43627.
1:: (   (𝐴𝐵𝐶𝐷)   ▶   (𝐴𝐵 𝐶𝐷)   )
2:: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥𝐵𝑦𝐷𝜑   )
3:1: (   (𝐴𝐵𝐶𝐷)   ▶   𝐴𝐵   )
4:3,2: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥𝐴𝑦𝐷𝜑   )
5:4: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥(𝑥𝐴 → ∀𝑦𝐷𝜑)   )
6:5: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   (𝑥𝐴 → ∀𝑦𝐷𝜑)   )
7:: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑, 𝑥𝐴   ▶   𝑥𝐴   )
8:7,6: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑, 𝑥𝐴   ▶   𝑦𝐷𝜑   )
9:1: (   (𝐴𝐵𝐶𝐷)   ▶   𝐶𝐷   )
10:9,8: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑, 𝑥𝐴   ▶   𝑦𝐶𝜑   )
11:10: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   (𝑥𝐴 → ∀𝑦𝐶𝜑)   )
12:: ((𝐴𝐵𝐶𝐷) → ∀𝑥(𝐴𝐵𝐶𝐷))
13:: (∀𝑥𝐵𝑦𝐷𝜑 → ∀𝑥𝑥𝐵𝑦𝐷𝜑)
14:12,13,11: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥(𝑥𝐴 → ∀𝑦𝐶𝜑)   )
15:14: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥𝐴𝑦𝐶𝜑   )
16:15: (   (𝐴𝐵𝐶𝐷)    ▶   (∀𝑥𝐵𝑦𝐷𝜑 → ∀𝑥𝐴𝑦𝐶𝜑)   )
qed:16: ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷𝜑 → ∀𝑥𝐴𝑦𝐶𝜑))
(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))
 
TheoremordelordALTVD 43628 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 9615. 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 43298 is ordelordALTVD 43628 without virtual deductions and was automatically derived from ordelordALTVD 43628 using the tools program translate..without..overwriting.cmd and the Metamath program "MM-PA> MINIMIZE_WITH *" command.
1:: (   (Ord 𝐴𝐵𝐴)   ▶   (Ord 𝐴 𝐵𝐴)   )
2:1: (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐴   )
3:1: (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
4:2: (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐴   )
5:2: (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴 𝑦𝐴(𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
6:4,3: (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
7:6,6,5: (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐵 𝑦𝐵(𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
8:: ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
9:8: 𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
10:9: 𝑦𝐴((𝑥𝑦𝑥 = 𝑦 𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
11:10: (∀𝑦𝐴(𝑥𝑦𝑥 = 𝑦 𝑦𝑥) ↔ ∀𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦))
12:11: 𝑥(∀𝑦𝐴(𝑥𝑦𝑥 = 𝑦 𝑦𝑥) ↔ ∀𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦))
13:12: 𝑥𝐴(∀𝑦𝐴(𝑥𝑦 𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦))
14:13: (∀𝑥𝐴𝑦𝐴(𝑥𝑦 𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥 𝑥 = 𝑦))
15:14,5: (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
16:4,15,3: (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐵   )
17:16,7: (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐵   )
qed:17: ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((Ord 𝐴𝐵𝐴) → Ord 𝐵)
 
TheoremequncomVD 43629 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 43629 without virtual deductions and was automatically derived from equncomVD 43629.
1:: (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐵𝐶)   )
2:: (𝐵𝐶) = (𝐶𝐵)
3:1,2: (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐶𝐵)   )
4:3: (𝐴 = (𝐵𝐶) → 𝐴 = (𝐶𝐵))
5:: (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐶𝐵)   )
6:5,2: (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐵𝐶)   )
7:6: (𝐴 = (𝐶𝐵) → 𝐴 = (𝐵𝐶))
8:4,7: (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
(Contributed by Alan Sare, 17-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
 
TheoremequncomiVD 43630 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 43630 without virtual deductions and was automatically derived from equncomiVD 43630.
h1:: 𝐴 = (𝐵𝐶)
qed:1: 𝐴 = (𝐶𝐵)
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 = (𝐵𝐶)       𝐴 = (𝐶𝐵)
 
TheoremsucidALTVD 43631 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 43632 is sucidALTVD 43631 without virtual deductions and was automatically derived from sucidALTVD 43631. 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 𝑥𝐴𝑦𝐴(𝑥𝑦𝑥 = 𝑦𝑦𝑥) is a semantic variation of the theorem (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑥 = 𝑦𝑦𝑥))), which unifies with the set.mm reference definition (axiom) dford2 9615.
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
 
TheoremsucidALT 43632 A set belongs to its successor. This proof was automatically derived from sucidALTVD 43631 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
 
TheoremsucidVD 43633 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 43633 without virtual deductions and was automatically derived from sucidVD 43633.
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
 
Theoremimbi12VD 43634 Implication form of imbi12i 351. 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 347 is imbi12VD 43634 without virtual deductions and was automatically derived from imbi12VD 43634.
1:: (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2:: (   (𝜑𝜓)   ,   (𝜒𝜃)    ▶   (𝜒𝜃)   )
3:: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜑 𝜒)   ▶   (𝜑𝜒)   )
4:1,3: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜑 𝜒)   ▶   (𝜓𝜒)   )
5:2,4: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜑 𝜒)   ▶   (𝜓𝜃)   )
6:5: (   (𝜑𝜓)   ,   (𝜒𝜃)    ▶   ((𝜑𝜒) → (𝜓𝜃))   )
7:: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜓 𝜃)   ▶   (𝜓𝜃)   )
8:1,7: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜓 𝜃)   ▶   (𝜑𝜃)   )
9:2,8: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜓 𝜃)   ▶   (𝜑𝜒)   )
10:9: (   (𝜑𝜓)   ,   (𝜒𝜃)    ▶   ((𝜓𝜃) → (𝜑𝜒))   )
11:6,10: (   (𝜑𝜓)   ,   (𝜒𝜃)    ▶   ((𝜑𝜒) ↔ (𝜓𝜃))   )
12:11: (   (𝜑𝜓)   ▶   ((𝜒𝜃) → ((𝜑𝜒) ↔ (𝜓𝜃)))   )
qed:12: ((𝜑𝜓) → ((𝜒𝜃) → ((𝜑𝜒) ↔ (𝜓𝜃))))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜃) → ((𝜑𝜒) ↔ (𝜓𝜃))))
 
Theoremimbi13VD 43635 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 43281 is imbi13VD 43635 without virtual deductions and was automatically derived from imbi13VD 43635.
1:: (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2:: (   (𝜑𝜓)   ,   (𝜒𝜃)    ▶   (𝜒𝜃)   )
3:: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏 𝜂)   ▶   (𝜏𝜂)   )
4:2,3: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏 𝜂)   ▶   ((𝜒𝜏) ↔ (𝜃𝜂))   )
5:1,4: (   (𝜑𝜓)   ,   (𝜒𝜃)   ,   (𝜏 𝜂)   ▶   ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))   )
6:5: (   (𝜑𝜓)   ,   (𝜒𝜃)    ▶   ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃 𝜂))))   )
7:6: (   (𝜑𝜓)   ▶   ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃 𝜂)))))   )
qed:7: ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃 𝜂))))))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))
 
Theoremsbcim2gVD 43636 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 43299 is sbcim2gVD 43636 without virtual deductions and was automatically derived from sbcim2gVD 43636.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (   𝐴𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓𝜒))   )
3:1,2: (   𝐴𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 𝜒))   ▶   ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))   )
4:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   )
5:3,4: (   𝐴𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒))   )
6:5: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒)))   )
7:: (   𝐴𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   )
8:4,7: (   𝐴𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 [𝐴 / 𝑥](𝜓𝜒))   )
9:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 𝜒)) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)))   )
10:8,9: (   𝐴𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 𝜒))   )
11:10: (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 𝜒)))   )
12:6,11: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒)))   )
qed:12: (𝐴𝐵 → ([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.)
(𝐴𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
 
TheoremsbcbiVD 43637 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 43300 is sbcbiVD 43637 without virtual deductions and was automatically derived from sbcbiVD 43637.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (   𝐴𝐵   ,   𝑥(𝜑𝜓)    ▶   𝑥(𝜑𝜓)   )
3:1,2: (   𝐴𝐵   ,   𝑥(𝜑𝜓)    ▶   [𝐴 / 𝑥](𝜑𝜓)   )
4:1,3: (   𝐴𝐵   ,   𝑥(𝜑𝜓)    ▶   ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)   )
5:4: (   𝐴𝐵   ▶   (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
qed:5: (𝐴𝐵 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 
TheoremtrsbcVD 43638* 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 43301 is trsbcVD 43638 without virtual deductions and was automatically derived from trsbcVD 43638.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑧𝑦 𝑧𝑦)   )
3:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝑥 𝑦𝐴)   )
4:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑧𝑥 𝑧𝐴)   )
5:1,2,3,4: (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑧𝑦 → ([𝐴 / 𝑥]𝑦𝑥[𝐴 / 𝑥]𝑧𝑥)) ↔ (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))   )
6:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ ([𝐴 / 𝑥]𝑧𝑦 ([𝐴 / 𝑥]𝑦𝑥[𝐴 / 𝑥]𝑧𝑥)))   )
7:5,6: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ (𝑧𝑦 → (𝑦𝐴 𝑧𝐴)))   )
8:: ((𝑧𝑦 → (𝑦𝐴 𝑧𝐴)) ↔ ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
9:7,8: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ ((𝑧𝑦𝑦𝐴) 𝑧𝐴))   )
10:: ((𝑧𝑦 → (𝑦𝑥 𝑧𝑥)) ↔ ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
11:10: 𝑥((𝑧𝑦 → (𝑦𝑥 𝑧𝑥)) ↔ ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
12:1,11: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ [𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) 𝑧𝑥))   )
13:9,12: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝑧𝑦 𝑦𝑥) → 𝑧𝑥) ↔ ((𝑧𝑦𝑦𝐴) 𝑧𝐴))   )
14:13: (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥]((𝑧 𝑦𝑦𝑥) → 𝑧𝑥) ↔ ((𝑧𝑦𝑦𝐴) 𝑧𝐴))   )
15:14: (   𝐴𝐵   ▶   (∀𝑦[𝐴 / 𝑥]((𝑧 𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦((𝑧𝑦𝑦𝐴) 𝑧𝐴))   )
16:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦((𝑧 𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧𝑦 𝑦𝑥) → 𝑧𝑥))   )
17:15,16: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦((𝑧 𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦((𝑧𝑦𝑦𝐴) 𝑧𝐴))   )
18:17: (   𝐴𝐵   ▶   𝑧([𝐴 / 𝑥]𝑦(( 𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦((𝑧𝑦𝑦𝐴) 𝑧𝐴))   )
19:18: (   𝐴𝐵   ▶   (∀𝑧[𝐴 / 𝑥]𝑦(( 𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧𝑦((𝑧𝑦 𝑦𝐴) → 𝑧𝐴))   )
20:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑧𝑦(( 𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧[𝐴 / 𝑥]𝑦((𝑧 𝑦𝑦𝑥) → 𝑧𝑥))   )
21:19,20: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑧𝑦(( 𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧𝑦((𝑧𝑦 𝑦𝐴) → 𝑧𝐴))   )
22:: (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦 𝑦𝐴) → 𝑧𝐴))
23:21,22: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑧𝑦(( 𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ Tr 𝐴)   )
24:: (Tr 𝑥 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 𝑥) → 𝑧𝑥))
25:24: 𝑥(Tr 𝑥 ↔ ∀𝑧𝑦((𝑧𝑦 𝑦𝑥) → 𝑧𝑥))
26:1,25: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]Tr 𝑥 [𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))   )
27:23,26: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴)   )
qed:27: (𝐴𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
 
TheoremtruniALTVD 43639* 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 43302 is truniALTVD 43639 without virtual deductions and was automatically derived from truniALTVD 43639.
1:: (   𝑥𝐴Tr 𝑥   ▶   𝑥𝐴 Tr 𝑥   )
2:: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   (𝑧𝑦𝑦 𝐴)   )
3:2: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑧𝑦   )
4:2: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑦 𝐴   )
5:4: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑞(𝑦𝑞𝑞𝐴)   )
6:: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   (𝑦𝑞𝑞𝐴)   )
7:6: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   𝑦𝑞   )
8:6: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   𝑞𝐴   )
9:1,8: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   )
10:8,9: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   Tr 𝑞   )
11:3,7,10: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   𝑧𝑞   )
12:11,8: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   𝑧 𝐴   )
13:12: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   ((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
14:13: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
15:14: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
16:5,15: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑧 𝐴   )
17:16: (   𝑥𝐴Tr 𝑥   ▶   ((𝑧𝑦 𝑦 𝐴) → 𝑧 𝐴)   )
18:17: (   𝑥𝐴Tr 𝑥    ▶   𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
19:18: (   𝑥𝐴Tr 𝑥   ▶   Tr 𝐴   )
qed:19: (∀𝑥𝐴Tr 𝑥 → Tr 𝐴)
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 
Theoremee33VD 43640 Non-virtual deduction form of e33 43495. 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 43282 is ee33VD 43640 without virtual deductions and was automatically derived from ee33VD 43640.
h1:: (𝜑 → (𝜓 → (𝜒𝜃)))
h2:: (𝜑 → (𝜓 → (𝜒𝜏)))
h3:: (𝜃 → (𝜏𝜂))
4:1,3: (𝜑 → (𝜓 → (𝜒 → (𝜏𝜂))))
5:4: (𝜏 → (𝜑 → (𝜓 → (𝜒𝜂))))
6:2,5: (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
7:6: (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 𝜂)))))
8:7: (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))
qed:8: (𝜑 → (𝜓 → (𝜒𝜂)))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜒𝜏)))    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))
 
TheoremtrintALTVD 43641* The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 43642. 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 43642 is trintALTVD 43641 without virtual deductions and was automatically derived from trintALTVD 43641.
1:: (   𝑥𝐴Tr 𝑥   ▶   𝑥𝐴Tr 𝑥   )
2:: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑧𝑦𝑦 𝐴)   )
3:2: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧𝑦   )
4:2: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑦 𝐴   )
5:4: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞𝐴𝑦𝑞   )
6:5: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑞𝐴𝑦𝑞)   )
7:: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   𝑞𝐴   )
8:7,6: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   𝑦𝑞   )
9:7,1: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   )
10:7,9: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   Tr 𝑞   )
11:10,3,8: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   𝑧𝑞   )
12:11: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑞𝐴𝑧𝑞)   )
13:12: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞(𝑞𝐴𝑧𝑞)   )
14:13: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞𝐴𝑧𝑞   )
15:3,14: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧 𝐴   )
16:15: (   𝑥𝐴Tr 𝑥   ▶   ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
17:16: (   𝑥𝐴Tr 𝑥   ▶   𝑧𝑦((𝑧 𝑦𝑦 𝐴) → 𝑧 𝐴)   )
18:17: (   𝑥𝐴Tr 𝑥   ▶   Tr 𝐴   )
qed:18: (∀𝑥𝐴Tr 𝑥 → Tr 𝐴)
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 
TheoremtrintALT 43642* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 43642 is an alternate proof of trint 5284. trintALT 43642 is trintALTVD 43641 without virtual deductions and was automatically derived from trintALTVD 43641 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.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 
Theoremundif3VD 43643 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 43643 without virtual deductions and was automatically derived from undif3VD 43643.
1:: (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴 𝑥 ∈ (𝐵𝐶)))
2:: (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥 𝐶))
3:2: ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4:1,3: (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
5:: (   𝑥𝐴   ▶   𝑥𝐴   )
6:5: (   𝑥𝐴   ▶   (𝑥𝐴𝑥𝐵)   )
7:5: (   𝑥𝐴   ▶   𝑥𝐶𝑥𝐴)   )
8:6,7: (   𝑥𝐴   ▶   ((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐶𝑥𝐴))   )
9:8: (𝑥𝐴 → ((𝑥𝐴𝑥𝐵) ∧ ( ¬ 𝑥𝐶𝑥𝐴)))
10:: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   )
11:10: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   𝑥𝐵   )
12:10: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   ¬ 𝑥𝐶    )
13:11: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 𝑥𝐵)   )
14:12: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   𝑥 𝐶𝑥𝐴)   )
15:13,14: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   ((𝑥 𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴))   )
16:15: ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → ((𝑥𝐴 𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
17:9,16: ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
18:: (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   )
19:18: (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   𝑥𝐴   )
20:18: (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   ¬ 𝑥𝐶    )
21:18: (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
22:21: ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
23:: (   (𝑥𝐴𝑥𝐴)   ▶   (𝑥𝐴 𝑥𝐴)   )
24:23: (   (𝑥𝐴𝑥𝐴)   ▶   𝑥𝐴   )
25:24: (   (𝑥𝐴𝑥𝐴)   ▶   (𝑥𝐴 (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
26:25: ((𝑥𝐴𝑥𝐴) → (𝑥𝐴 ∨ ( 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
27:10: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
28:27: ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
29:: (   (𝑥𝐵𝑥𝐴)   ▶   (𝑥𝐵 𝑥𝐴)   )
30:29: (   (𝑥𝐵𝑥𝐴)   ▶   𝑥𝐴   )
31:30: (   (𝑥𝐵𝑥𝐴)   ▶   (𝑥𝐴 (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
32:31: ((𝑥𝐵𝑥𝐴) → (𝑥𝐴 ∨ ( 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
33:22,26: (((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐴 𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
34:28,32: (((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵 𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
35:33,34: ((((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥 𝐴𝑥𝐴)) ∨ ((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵𝑥𝐴))) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
36:: ((((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥 𝐴𝑥𝐴)) ∨ ((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵𝑥𝐴))) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
37:36,35: (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶 𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
38:17,37: ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
39:: (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥 𝐴))
40:39: 𝑥 ∈ (𝐶𝐴) ↔ ¬ (𝑥𝐶 ¬ 𝑥𝐴))
41:: (¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴) ↔ (¬ 𝑥 𝐶𝑥𝐴))
42:40,41: 𝑥 ∈ (𝐶𝐴) ↔ (¬ 𝑥𝐶𝑥 𝐴))
43:: (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵 ))
44:43,42: ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴) ) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
45:: (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ ( 𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
46:45,44: (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ ( (𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
47:4,38: (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ ((𝑥𝐴 𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
48:46,47: (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴 𝐵) ∖ (𝐶𝐴)))
49:48: 𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ((𝐴𝐵) ∖ (𝐶𝐴)))
qed:49: (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶 𝐴))
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴))
 
TheoremsbcssgVD 43644 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 43644 without virtual deductions and was automatically derived from sbcssgVD 43644.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦 𝐴 / 𝑥𝐶)   )
3:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐷𝑦 𝐴 / 𝑥𝐷)   )
4:2,3: (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑦𝐶 [𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷 ))   )
5:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶 𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))   )
6:4,5: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶 𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
7:6: (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥](𝑦 𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
8:7: (   𝐴𝐵   ▶   (∀𝑦[𝐴 / 𝑥](𝑦 𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷) )   )
9:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦(𝑦 𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷))   )
10:8,9: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦(𝑦 𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷) )   )
11:: (𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷))
110:11: 𝑥(𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦 𝐷))
12:1,110: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷 [𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷))   )
13:10,12: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷 𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
14:: (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀ 𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))
15:13,14: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷 𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)   )
qed:15: (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷 𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
 
TheoremcsbingVD 43645 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 43645 without virtual deductions and was automatically derived from csbingVD 43645.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷) }
20:2: 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦 𝐷)}
30:1,20: (   𝐴𝐵   ▶   [𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
3:1,30: (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
4:1: (   𝐴𝐵   ▶   𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶 𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
5:3,4: (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
6:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦 𝐴 / 𝑥𝐶)   )
7:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐷𝑦 𝐴 / 𝑥𝐷)   )
8:6,7: (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑦𝐶 [𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷 ))   )
9:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶 𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))   )
10:9,8: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶 𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
11:10: (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥](𝑦 𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
12:11: (   𝐴𝐵   ▶   {𝑦[𝐴 / 𝑥](𝑦𝐶 𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
13:5,12: (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
14:: (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = { 𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}
15:13,14: (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)   )
qed:15: (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = ( 𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
 
TheoremonfrALTlem5VD 43646* Virtual deduction proof of onfrALTlem5 43303. 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 43303 is onfrALTlem5VD 43646 without virtual deductions and was automatically derived from onfrALTlem5VD 43646.
1:: 𝑎 ∈ V
2:1: (𝑎𝑥) ∈ V
3:2: ([(𝑎𝑥) / 𝑏]𝑏 = ∅ ↔ (𝑎 𝑥) = ∅)
4:3: [(𝑎𝑥) / 𝑏]𝑏 = ∅ ↔ ¬ (𝑎𝑥) = ∅)
5:: ((𝑎𝑥) ≠ ∅ ↔ ¬ (𝑎𝑥 ) = ∅)
6:4,5: [(𝑎𝑥) / 𝑏]𝑏 = ∅ ↔ (𝑎𝑥) ≠ ∅)
7:2: [(𝑎𝑥) / 𝑏]𝑏 = ∅ ↔ [(𝑎𝑥) / 𝑏]¬ 𝑏 = ∅)
8:: (𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅)
9:8: 𝑏(𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅)
10:2,9: ([(𝑎𝑥) / 𝑏]𝑏 ≠ ∅ ↔ [(𝑎𝑥) / 𝑏]¬ 𝑏 = ∅)
11:7,10: [(𝑎𝑥) / 𝑏]𝑏 = ∅ ↔ [(𝑎𝑥) / 𝑏]𝑏 ≠ ∅)
12:6,11: ([(𝑎𝑥) / 𝑏]𝑏 ≠ ∅ ↔ ( 𝑎𝑥) ≠ ∅)
13:2: ([(𝑎𝑥) / 𝑏]𝑏 ⊆ (𝑎𝑥 ) ↔ (𝑎𝑥) ⊆ (𝑎𝑥))
14:12,13: (([(𝑎𝑥) / 𝑏]𝑏 ⊆ (𝑎 𝑥) ∧ [(𝑎𝑥) / 𝑏]𝑏 ≠ ∅) ↔ ((𝑎𝑥) ⊆ (𝑎 𝑥) ∧ (𝑎𝑥) ≠ ∅))
15:2: ([(𝑎𝑥) / 𝑏](𝑏 ⊆ (𝑎 𝑥) ∧ 𝑏 ≠ ∅) ↔ ([(𝑎𝑥) / 𝑏]𝑏 ⊆ (𝑎𝑥) ∧ [(𝑎𝑥) / 𝑏]𝑏 ≠ ∅))
16:15,14: ([(𝑎𝑥) / 𝑏](𝑏 ⊆ (𝑎 𝑥) ∧ 𝑏 ≠ ∅) ↔ ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅))
17:2: (𝑎𝑥) / 𝑏(𝑏𝑦) = ( (𝑎𝑥) / 𝑏𝑏(𝑎𝑥) / 𝑏𝑦)
18:2: (𝑎𝑥) / 𝑏𝑏 = (𝑎𝑥)
19:2: (𝑎𝑥) / 𝑏𝑦 = 𝑦
20:18,19: ((𝑎𝑥) / 𝑏𝑏(𝑎 𝑥) / 𝑏𝑦) = ((𝑎𝑥) ∩ 𝑦)
21:17,20: (𝑎𝑥) / 𝑏(𝑏𝑦) = (( 𝑎𝑥) ∩ 𝑦)
22:2: ([(𝑎𝑥) / 𝑏](𝑏𝑦) = ∅ ↔ (𝑎𝑥) / 𝑏(𝑏𝑦) = (𝑎𝑥) / 𝑏 ∅)
23:2: (𝑎𝑥) / 𝑏∅ = ∅
24:21,23: ((𝑎𝑥) / 𝑏(𝑏𝑦) = (𝑎𝑥) / 𝑏∅ ↔ ((𝑎𝑥) ∩ 𝑦) = ∅)
25:22,24: ([(𝑎𝑥) / 𝑏](𝑏𝑦) = ∅ ↔ ((𝑎𝑥) ∩ 𝑦) = ∅)
26:2: ([(𝑎𝑥) / 𝑏]𝑦𝑏𝑦 (𝑎𝑥))
27:25,26: (([(𝑎𝑥) / 𝑏]𝑦𝑏[ (𝑎𝑥) / 𝑏](𝑏𝑦) = ∅) ↔ (𝑦 ∈ (𝑎𝑥) ∧ (( 𝑎𝑥) ∩ 𝑦) = ∅))
28:2: ([(𝑎𝑥) / 𝑏](𝑦𝑏 ∧ (𝑏 𝑦) = ∅) ↔ ([(𝑎𝑥) / 𝑏]𝑦𝑏[(𝑎𝑥) / 𝑏](𝑏𝑦) = ∅))
29:27,28: ([(𝑎𝑥) / 𝑏](𝑦𝑏 ∧ (𝑏 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅))
30:29: 𝑦([(𝑎𝑥) / 𝑏](𝑦𝑏 ∧ (𝑏𝑦) = ∅) ↔ (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅))
31:30: (∃𝑦[(𝑎𝑥) / 𝑏](𝑦𝑏 ∧ (𝑏𝑦) = ∅) ↔ ∃𝑦(𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) 𝑦) = ∅))
32:: (∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅ ))
33:31,32: (∃𝑦[(𝑎𝑥) / 𝑏](𝑦𝑏 ∧ (𝑏𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅)
34:2: (∃𝑦[(𝑎𝑥) / 𝑏](𝑦𝑏 ∧ (𝑏𝑦) = ∅) ↔ [(𝑎𝑥) / 𝑏]𝑦(𝑦𝑏 ∧ ( 𝑏𝑦) = ∅))
35:33,34: ([(𝑎𝑥) / 𝑏]𝑦(𝑦𝑏 ∧ (𝑏𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦 ) = ∅)
36:: (∃𝑦𝑏(𝑏𝑦) = ∅ ↔ ∃𝑦 (𝑦𝑏 ∧ (𝑏𝑦) = ∅))
37:36: 𝑏(∃𝑦𝑏(𝑏𝑦) = ∅ ↔ 𝑦(𝑦𝑏 ∧ (𝑏𝑦) = ∅))
38:2,37: ([(𝑎𝑥) / 𝑏]𝑦𝑏(𝑏 𝑦) = ∅ ↔ [(𝑎𝑥) / 𝑏]𝑦(𝑦𝑏 ∧ (𝑏𝑦) = ∅))
39:35,38: ([(𝑎𝑥) / 𝑏]𝑦𝑏(𝑏 𝑦) = ∅ ↔ ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅)
40:16,39: (([(𝑎𝑥) / 𝑏](𝑏 ⊆ (𝑎 𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎𝑥) / 𝑏]𝑦𝑏(𝑏 𝑦) = ∅) ↔ (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
41:2: ([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏(𝑏𝑦) = ∅) ↔ ([(𝑎 𝑥) / 𝑏](𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎𝑥) / 𝑏]𝑦𝑏(𝑏𝑦) = ∅))
qed:40,41: ([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏(𝑏𝑦) = ∅) ↔ (((𝑎 𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥 )((𝑎𝑥) ∩ 𝑦) = ∅))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅) ↔ (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
 
TheoremonfrALTlem4VD 43647* Virtual deduction proof of onfrALTlem4 43304. 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 43304 is onfrALTlem4VD 43647 without virtual deductions and was automatically derived from onfrALTlem4VD 43647.
1:: 𝑦 ∈ V
2:1: ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ 𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅)
3:1: 𝑦 / 𝑥(𝑎𝑥) = (𝑦 / 𝑥 𝑎𝑦 / 𝑥𝑥)
4:1: 𝑦 / 𝑥𝑎 = 𝑎
5:1: 𝑦 / 𝑥𝑥 = 𝑦
6:4,5: (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥) = ( 𝑎𝑦)
7:3,6: 𝑦 / 𝑥(𝑎𝑥) = (𝑎𝑦)
8:1: 𝑦 / 𝑥∅ = ∅
9:7,8: (𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥 ∅ ↔ (𝑎𝑦) = ∅)
10:2,9: ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ (𝑎 𝑦) = ∅)
11:1: ([𝑦 / 𝑥]𝑥𝑎𝑦𝑎)
12:11,10: (([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥]( 𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
13:1: ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅))
qed:13,12: ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
 
TheoremonfrALTlem3VD 43648* Virtual deduction proof of onfrALTlem3 43305. 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 43305 is onfrALTlem3VD 43648 without virtual deductions and was automatically derived from onfrALTlem3VD 43648.
1:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
2:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   )
3:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑥𝑎   )
4:1: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 On   )
5:3,4: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑥 ∈ On   )
6:5: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   Ord 𝑥   )
7:6: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    E We 𝑥   )
8:: (𝑎𝑥) ⊆ 𝑥
9:7,8: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    E We (𝑎𝑥)   )
10:9: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    E Fr (𝑎𝑥)   )
11:10: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ∅) → ∃𝑦𝑏(𝑏𝑦) = ∅)   )
12:: 𝑥 ∈ V
13:12,8: (𝑎𝑥) ∈ V
14:13,11: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   [(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏(𝑏𝑦) = ∅)   )
15:: ([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏(𝑏𝑦) = ∅) ↔ (((𝑎 𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)( (𝑎𝑥) ∩ 𝑦) = ∅))
16:14,15: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ ( 𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅)   )
17:: (𝑎𝑥) ⊆ (𝑎𝑥)
18:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   ¬ (𝑎𝑥) = ∅   )
19:18: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (𝑎𝑥) ≠ ∅   )
20:17,19: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎 𝑥) ≠ ∅)   )
qed:16,20: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦 ) = ∅   )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅   )
 
Theoremsimplbi2comtVD 43649 Virtual deduction proof of simplbi2comt 503. 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 503 is simplbi2comtVD 43649 without virtual deductions and was automatically derived from simplbi2comtVD 43649.
1:: (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜑 ↔ ( 𝜓𝜒))   )
2:1: (   (𝜑 ↔ (𝜓𝜒))   ▶   ((𝜓𝜒 ) → 𝜑)   )
3:2: (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜓 → (𝜒 𝜑))   )
4:3: (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜒 → (𝜓 𝜑))   )
qed:4: ((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓 𝜑)))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))
 
TheoremonfrALTlem2VD 43650* Virtual deduction proof of onfrALTlem2 43307. 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 43307 is onfrALTlem2VD 43650 without virtual deductions and was automatically derived from onfrALTlem2VD 43650.
1:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   ▶   ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   )
2:1: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   ▶   𝑧 ∈ (𝑎𝑦)   )
3:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   ▶   𝑧𝑎   )
4:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
5:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   )
6:5: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑥𝑎   )
7:4: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 On   )
8:6,7: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑥 ∈ On   )
9:8: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   Ord 𝑥   )
10:9: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   Tr 𝑥   )
11:1: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   ▶   𝑦 ∈ (𝑎𝑥)   )
12:11: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   ▶   𝑦𝑥   )
13:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   ▶   𝑧𝑦   )
14:10,12,13: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   ▶   𝑧𝑥   )
15:3,14: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   ▶   𝑧 ∈ (𝑎𝑥)   )
16:13,15: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦))   ▶   𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)   )
17:16: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑧 ∈ (𝑎𝑦) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦))   )
18:17: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦 ) = ∅)   ▶   𝑧(𝑧 ∈ (𝑎𝑦) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦))   )
19:18: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑎𝑦) ⊆ ((𝑎𝑥) ∩ 𝑦)   )
20:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅)   )
21:20: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦 ) = ∅)   ▶   ((𝑎𝑥) ∩ 𝑦) = ∅   )
22:19,21: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑎𝑦) = ∅   )
23:20: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦 ) = ∅)   ▶   𝑦 ∈ (𝑎𝑥)   )
24:23: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦 ) = ∅)   ▶   𝑦𝑎   )
25:22,24: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅), (𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑦𝑎 ∧ (𝑎𝑦) = ∅)   )
26:25: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) 𝑦) = ∅) → (𝑦𝑎 ∧ (𝑎𝑦) = ∅))   )
27:26: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥 ) ∩ 𝑦) = ∅) → (𝑦𝑎 ∧ (𝑎𝑦) = ∅))   )
28:27: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (∃𝑦(𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥 ) ∩ 𝑦) = ∅) → ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅))   )
29:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦 ) = ∅   )
30:29: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦(𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) 𝑦) = ∅)   )
31:28,30: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅)   )
qed:31: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦𝑎(𝑎𝑦) = ∅   )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
 
TheoremonfrALTlem1VD 43651* Virtual deduction proof of onfrALTlem1 43309. 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 43309 is onfrALTlem1VD 43651 without virtual deductions and was automatically derived from onfrALTlem1VD 43651.
1:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   )
2:1: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   𝑥(𝑥𝑎 ∧ (𝑎𝑥) = ∅)   )
3:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅)    )
4:: ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅ ) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
5:4: 𝑦([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
6:5: (∃𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅))
7:3,6: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅)   )
8:: (∃𝑦𝑎(𝑎𝑦) = ∅ ↔ ∃𝑦( 𝑦𝑎 ∧ (𝑎𝑦) = ∅))
qed:7,8: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   𝑦𝑎(𝑎𝑦) = ∅   )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
 
TheoremonfrALTVD 43652 Virtual deduction proof of onfrALT 43310. 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 43310 is onfrALTVD 43652 without virtual deductions and was automatically derived from onfrALTVD 43652.
1:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦𝑎(𝑎𝑦) = ∅   )
2:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦𝑎(𝑎𝑦) = ∅   )
3:1: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶    (¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
4:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶    ((𝑎𝑥) = ∅ → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
5:: ((𝑎𝑥) = ∅ ∨ ¬ (𝑎𝑥) = ∅)
6:5,4,3: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶    𝑦𝑎(𝑎𝑦) = ∅   )
7:6: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥𝑎 → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
8:7: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑥(𝑥 𝑎 → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
9:8: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥𝑥 𝑎 → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
10:: (𝑎 ≠ ∅ ↔ ∃𝑥𝑥𝑎)
11:9,10: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ∅ → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
12:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 On ∧ 𝑎 ≠ ∅)   )
13:12: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎    )
14:13,11: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑦 𝑎(𝑎𝑦) = ∅   )
15:14: ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
16:15: 𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 𝑎(𝑎𝑦) = ∅)
qed:16: E Fr On
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
E Fr On
 
Theoremcsbeq2gVD 43653 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 43653 without virtual deductions and was automatically derived from csbeq2gVD 43653.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   (∀𝑥𝐵 = 𝐶[𝐴 / 𝑥] 𝐵 = 𝐶)   )
3:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)   )
4:2,3: (   𝐴𝑉   ▶   (∀𝑥𝐵 = 𝐶𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐶)   )
qed:4: (𝐴𝑉 → (∀𝑥𝐵 = 𝐶𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐶))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
 
TheoremcsbsngVD 43654 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 43654 without virtual deductions and was automatically derived from csbsngVD 43654.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵)   )
3:1: (   𝐴𝑉   ▶   𝐴 / 𝑥𝑦 = 𝑦   )
4:3: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
5:2,4: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 𝑦 = 𝐴 / 𝑥𝐵)   )
6:5: (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
7:6: (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
8:1: (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
9:7,8: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
10:: {𝐵} = {𝑦𝑦 = 𝐵}
11:10: 𝑥{𝐵} = {𝑦𝑦 = 𝐵}
12:1,11: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
13:9,12: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = { 𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
14:: {𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}
15:13,14: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = { 𝐴 / 𝑥𝐵}   )
qed:15: (𝐴𝑉𝐴 / 𝑥{𝐵} = { 𝐴 / 𝑥𝐵})
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
 
TheoremcsbxpgVD 43655 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 43655 without virtual deductions and was automatically derived from csbxpgVD 43655.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝐵 𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵)   )
3:1: (   𝐴𝑉   ▶   𝐴 / 𝑥𝑤 = 𝑤   )
4:3: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵𝑤𝐴 / 𝑥𝐵)   )
5:2,4: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝐵𝑤 𝐴 / 𝑥𝐵)   )
6:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐶 𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶)   )
7:1: (   𝐴𝑉   ▶   𝐴 / 𝑥𝑦 = 𝑦   )
8:7: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐶)   )
9:6,8: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦 𝐴 / 𝑥𝐶)   )
10:5,9: (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑤𝐵 [𝐴 / 𝑥]𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵 𝑦𝐴 / 𝑥𝐶))   )
11:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑤𝐵 𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶))   )
12:10,11: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑤𝐵 𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))   )
13:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑧 = ⟨𝑤   ,    𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩)   )
14:12,13: (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦 ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
15:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦 [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)))   )
16:14,15: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
17:16: (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥](𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
18:17: (   𝐴𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
19:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))   )
20:18,19: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
21:20: (   𝐴𝑉   ▶   𝑤([𝐴 / 𝑥]𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
22:21: (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
23:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]𝑦 (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))   )
24:22,23: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
25:24: (   𝐴𝑉   ▶   𝑧([𝐴 / 𝑥]𝑤 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
26:25: (   𝐴𝑉   ▶   {𝑧[𝐴 / 𝑥]𝑤 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}    )
27:1: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑤 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧[𝐴 / 𝑥] 𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}   )
28:26,27: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑤 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}    )
29:: {⟨𝑤   ,   𝑦⟩ ∣ (𝑤𝐵𝑦𝐶)} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
30:: (𝐵 × 𝐶) = {⟨𝑤   ,   𝑦⟩ ∣ (𝑤𝐵 𝑦𝐶)}
31:29,30: (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤 , 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
32:31: 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
33:1,32: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 × 𝐶) = 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵 𝑦𝐶))}   )
34:28,33: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵 𝑦𝐴 / 𝑥𝐶))}   )
35:: {⟨𝑤   ,   𝑦⟩ ∣ (𝑤𝐴 / 𝑥𝐵 𝑦𝐴 / 𝑥𝐶)} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}
36:: (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = { 𝑤, 𝑦⟩ ∣ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)}
37:35,36: (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵 𝑦𝐴 / 𝑥𝐶))}
38:34,37: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶)   )
qed:38: (𝐴𝑉𝐴 / 𝑥(𝐵 × 𝐶) = ( 𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))
 
TheoremcsbresgVD 43656 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 43656 without virtual deductions and was automatically derived from csbresgVD 43656.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   𝐴 / 𝑥V = V   )
3:2: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V)   )
4:1: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)   )
5:3,4: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)   )
6:5: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
7:1: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))   )
8:6,7: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
9:: (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
10:9: 𝑥(𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
11:1,10: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))   )
12:8,11: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = ( 𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
13:: (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = ( 𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
14:12,13: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = ( 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)   )
qed:14: (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = ( 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
TheoremcsbrngVD 43657 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 43657 without virtual deductions and was automatically derived from csbrngVD 43657.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤   ,   𝑦 𝐵𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
3:1: (   𝐴𝑉   ▶   𝐴 / 𝑥𝑤   ,   𝑦⟩ = 𝑤, 𝑦   )
4:3: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑤   ,   𝑦 𝐴 / 𝑥𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
5:2,4: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤   ,   𝑦 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
6:5: (   𝐴𝑉   ▶   𝑤([𝐴 / 𝑥]𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
7:6: (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
8:1: (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑤   ,    𝑦⟩ ∈ 𝐵[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵)   )
9:7,8: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑤    ,   𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
10:9: (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥]𝑤 𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
11:10: (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑤 𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
12:1: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦 ∣ ∃𝑤 𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵}   )
13:11,12: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦 ∣ ∃𝑤 𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
14:: ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤   ,   𝑦⟩ ∈ 𝐵}
15:14: 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤   ,   𝑦 𝐵}
16:1,15: (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}   )
17:13,16: (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = {𝑦 𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
18:: ran 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑤    ,   𝑦⟩ ∈ 𝐴 / 𝑥𝐵}
19:17,18: (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵   )
qed:19: (𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)
 
Theoremcsbima12gALTVD 43658 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 43658 without virtual deductions and was automatically derived from csbima12gALTVD 43658.
1:: (   𝐴𝐶   ▶   𝐴𝐶   )
2:1: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = ( 𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
3:2: (   𝐴𝐶   ▶    ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
4:1: (   𝐴𝐶   ▶    𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵)   )
5:3,4: (   𝐴𝐶   ▶    𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
6:: (𝐹𝐵) = ran (𝐹𝐵)
7:6: 𝑥(𝐹𝐵) = ran (𝐹𝐵)
8:1,7: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵)   )
9:5,8: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
10:: (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
11:9,10: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = ( 𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
qed:11: (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = ( 𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
 
TheoremcsbunigVD 43659 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 43659 without virtual deductions and was automatically derived from csbunigVD 43659.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑧𝑦𝑧 𝑦)   )
3:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐵𝑦 𝐴 / 𝑥𝐵)   )
4:2,3: (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑧𝑦 [𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
5:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦 𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵))   )
6:4,5: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦 𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
7:6: (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥](𝑧 𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
8:7: (   𝐴𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
9:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵))   )
10:8,9: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
11:10: (   𝐴𝑉   ▶   𝑧([𝐴 / 𝑥]𝑦( 𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
12:11: (   𝐴𝑉   ▶   {𝑧[𝐴 / 𝑥]𝑦( 𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦 𝑦𝐴 / 𝑥𝐵)}   )
13:1: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧 𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)}    )
14:12,13: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧 𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦 𝑦𝐴 / 𝑥𝐵)}   )
15:: 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
16:15: 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦 𝐵)}
17:1,16: (   𝐴𝑉   ▶   [𝐴 / 𝑥] 𝐵 = {𝑧 𝑦(𝑧𝑦𝑦𝐵)}   )
18:1,17: (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}   )
19:14,18: (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = {𝑧 𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}   )
20:: 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦 𝑦𝐴 / 𝑥𝐵)}
21:19,20: (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵   )
qed:21: (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
 
Theoremcsbfv12gALTVD 43660 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 43660 without virtual deductions and was automatically derived from csbfv12gALTVD 43660.
1:: (   𝐴𝐶   ▶   𝐴𝐶   )
2:1: (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦} = { 𝑦}   )
3:1: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹 “ {𝐵 }) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵})   )
4:1: (   𝐴𝐶   ▶   𝐴 / 𝑥{𝐵} = { 𝐴 / 𝑥𝐵}   )
5:4: (   𝐴𝐶   ▶   (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})   )
6:3,5: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹 “ {𝐵 }) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})   )
7:1: (   𝐴𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ 𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦})   )
8:6,2: (   𝐴𝐶   ▶   (𝐴 / 𝑥(𝐹 “ { 𝐵}) = 𝐴 / 𝑥{𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})   )
9:7,8: (   𝐴𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})    )
10:9: (   𝐴𝐶   ▶   𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})   )
11:10: (   𝐴𝐶   ▶   {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
12:1: (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}   )
13:11,12: (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦 }}   )
14:13: (   𝐴𝐶   ▶    𝐴 / 𝑥{𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 {𝐴 / 𝑥𝐵}) = {𝑦}}   )
15:1: (   𝐴𝐶   ▶   𝐴 / 𝑥 {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
16:14,15: (   𝐴𝐶   ▶   𝐴 / 𝑥 {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
17:: (𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
18:17: 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵 }) = {𝑦}}
19:1,18: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
20:16,19: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
21:: (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}
22:20,21: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
qed:22: (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
 
Theoremcon5VD 43661 Virtual deduction proof of con5 43283. 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 43283 is con5VD 43661 without virtual deductions and was automatically derived from con5VD 43661.
1:: (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2:1: (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜓𝜑)   )
3:2: (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑 → ¬ ¬ 𝜓 )   )
4:: (𝜓 ↔ ¬ ¬ 𝜓)
5:3,4: (   (𝜑 ↔ ¬ 𝜓)   ▶   𝜑𝜓)   )
qed:5: ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))
 
TheoremrelopabVD 43662 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 43662 without virtual deductions and was automatically derived from relopabVD 43662.
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 = 𝑦𝑦 = 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 = 𝑥𝑥 = 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⟩ → ∃𝑢𝑣𝑧 = ⟨ 𝑢, 𝑣⟩)
19:18: (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑦𝑢 𝑣𝑧 = ⟨𝑢, 𝑣⟩)
20:: (∃𝑦𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → 𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
21:19,20: (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
22:21: (∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑥 𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
23:: (∃𝑥𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → 𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
24:22,23: (∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢 𝑣𝑧 = ⟨𝑢, 𝑣⟩)
25:24: {𝑧 ∣ ∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢𝑣𝑧 = ⟨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: (∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑥 𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
32:31: {𝑧 ∣ ∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} = { 𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
320:25,32: {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣𝑧 = ⟨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: (∃𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑢 𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
39:38: {𝑧 ∣ ∃𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩} = { 𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
40:320,39: {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
41:: {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) }
42:: {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = {𝑧 ∣ ∃𝑢𝑣(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}
 
Theorem19.41rgVD 43663 Virtual deduction proof of 19.41rg 43311. 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 43311 is 19.41rgVD 43663 without virtual deductions and was automatically derived from 19.41rgVD 43663. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (𝜓 → (𝜑 → (𝜑𝜓)))
2:1: ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → ( 𝜑𝜓))))
3:2: 𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
4:3: (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 𝑥(𝜑 → (𝜑𝜓))))
5:: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   𝑥(𝜓 → ∀𝑥𝜓)   )
6:4,5: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))   )
7:: (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶    𝑥𝜓   )
8:6,7: (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶    𝑥(𝜑 → (𝜑𝜓))   )
9:8: (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶    (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))   )
10:9: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
11:5: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀ 𝑥𝜓)   )
12:10,11: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ( 𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
13:12: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓)))   )
14:13: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥 𝜑𝜓) → ∃𝑥(𝜑𝜓))   )
qed:14: (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 𝜓) → ∃𝑥(𝜑𝜓)))
(∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
 
Theorem2pm13.193VD 43664 Virtual deduction proof of 2pm13.193 43313. 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 43313 is 2pm13.193VD 43664 without virtual deductions and was automatically derived from 2pm13.193VD 43664. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2:1: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
3:2: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   )
4:1: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
5:3,4: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
6:5: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
7:6: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
8:2: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   )
9:7,8: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
10:9: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
11:10: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝜑   )
12:2,11: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
13:12: (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
14:: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
15:14: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
16:15: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
17:14: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑    )
18:16,17: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ( 𝜑𝑦 = 𝑣)   )
19:18: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
20:15: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
21:19: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
22:20,21: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
23:22: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
24:23: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
25:15,24: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
26:25: (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
qed:13,26: (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
(((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
 
TheoremhbimpgVD 43665 Virtual deduction proof of hbimpg 43315. 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 43315 is hbimpgVD 43665 without virtual deductions and was automatically derived from hbimpgVD 43665. (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))   ▶   𝑥𝜑 → ∀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 → (𝜑𝜓))
8:7: (∀𝑥¬ 𝜑 → ∀𝑥(𝜑𝜓))
9:6,8: (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)), ¬ 𝜑   ▶   𝑥(𝜑𝜓)   )
10:9: (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   𝜑 → ∀𝑥(𝜑𝜓))   )
11:: (𝜓 → (𝜑𝜓))
12:11: (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
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(𝜑𝜓))   )
16:10,15: (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ((¬ 𝜑𝜓) → ∀𝑥(𝜑𝜓))   )
17:: ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
18:16,17: (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ((𝜑𝜓) → ∀𝑥(𝜑𝜓))   )
19:: (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑥( 𝜑 → ∀𝑥𝜑))
20:: (∀𝑥(𝜓 → ∀𝑥𝜓) → ∀𝑥𝑥( 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((𝜑𝜓) → ∀𝑥(𝜑𝜓))   )
qed:22: ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑𝜓) → ∀𝑥(𝜑𝜓)))
((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑𝜓) → ∀𝑥(𝜑𝜓)))
 
TheoremhbalgVD 43666 Virtual deduction proof of hbalg 43316. 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 43316 is hbalgVD 43666 without virtual deductions and was automatically derived from hbalgVD 43666. (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 → ∀𝑦𝑥𝜑)   )
3:: (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
4:2,3: (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥𝑦𝜑)   )
5:: (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦( 𝜑 → ∀𝑥𝜑))
6:5,4: (   𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀ 𝑦𝜑 → ∀𝑥𝑦𝜑)   )
qed:6: (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦 𝜑 → ∀𝑥𝑦𝜑))
(∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))
 
TheoremhbexgVD 43667 Virtual deduction proof of hbexg 43317. 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 43317 is hbexgVD 43667 without virtual deductions and was automatically derived from hbexgVD 43667. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦(𝜑 → ∀𝑥𝜑)   )
2:1: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 𝑥(𝜑 → ∀𝑥𝜑)   )
3:2: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 (𝜑 → ∀𝑥𝜑)   )
4:3: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝜑 → ∀𝑥¬ 𝜑)   )
5:: (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦 𝑥(𝜑 → ∀𝑥𝜑))
6:: (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦 𝑦𝑥(𝜑 → ∀𝑥𝜑))
7:5: (∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ 𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
8:5,6,7: (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦 𝑥𝑦(𝜑 → ∀𝑥𝜑))
9:8,4: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 𝑥𝜑 → ∀𝑥¬ 𝜑)   )
10:9: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦𝜑 → ∀𝑥¬ 𝜑)   )
11:10: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 𝜑 → ∀𝑥¬ 𝜑)   )
12:11: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 (∀𝑦¬ 𝜑 → ∀𝑥𝑦¬ 𝜑)   )
13:12: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀ 𝑦¬ 𝜑 → ∀𝑥𝑦¬ 𝜑)   )
14:: (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥 𝑥𝑦(𝜑 → ∀𝑥𝜑))
15:13,14: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 (∀𝑦¬ 𝜑 → ∀𝑥𝑦¬ 𝜑)   )
16:15: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 (¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
17:16: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶    𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
18:: (∃𝑦𝜑 ↔ ¬ ∀𝑦¬ 𝜑)
19:17,18: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
20:18: (∀𝑥𝑦𝜑 ↔ ∀𝑥¬ ∀𝑦¬ 𝜑)
21:19,20: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥𝑦𝜑)   )
22:8,21: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
23:14,22: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
qed:23: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
(∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
 
Theoremax6e2eqVD 43668* The following User's Proof is a Virtual Deduction proof (see wvd1 43330) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2eq 43318 is ax6e2eqVD 43668 without virtual deductions and was automatically derived from ax6e2eqVD 43668. (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 = 𝑢𝑦 = 𝑢)   )
6:5: (   𝑥𝑥 = 𝑦   ▶   (𝑥 = 𝑢 → (𝑥 = 𝑢 𝑦 = 𝑢))   )
7:6: (∀𝑥𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢)))
8:7: (∀𝑥𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → ( 𝑥 = 𝑢𝑦 = 𝑢)))
9:: (∀𝑥𝑥 = 𝑦 ↔ ∀𝑥𝑥𝑥 = 𝑦)
10:8,9: (∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢 𝑦 = 𝑢)))
11:1,10: (   𝑥𝑥 = 𝑦   ▶   𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢𝑦 = 𝑢))   )
12:11: (   𝑥𝑥 = 𝑦   ▶   (∃𝑥𝑥 = 𝑢 → ∃𝑥 (𝑥 = 𝑢𝑦 = 𝑢))   )
13:: 𝑥𝑥 = 𝑢
14:13,12: (   𝑥𝑥 = 𝑦   ▶   𝑥(𝑥 = 𝑢𝑦 = 𝑢 )   )
140:14: (∀𝑥𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑢) )
141:140: (∀𝑥𝑥 = 𝑦 → ∀𝑥𝑥(𝑥 = 𝑢𝑦 = 𝑢))
15:1,141: (   𝑥𝑥 = 𝑦   ▶   𝑥𝑥(𝑥 = 𝑢 𝑦 = 𝑢)   )
16:1,15: (   𝑥𝑥 = 𝑦   ▶   𝑦𝑥(𝑥 = 𝑢 𝑦 = 𝑢)   )
17:16: (   𝑥𝑥 = 𝑦   ▶   𝑦𝑥(𝑥 = 𝑢 𝑦 = 𝑢)   )
18:17: (   𝑥𝑥 = 𝑦   ▶   𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑢)   )
19:: (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   )
20:: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   (𝑥 = 𝑢𝑦 = 𝑢)   )
21:20: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑦 = 𝑢    )
22:19,21: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑦 = 𝑣    )
23:20: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   𝑥 = 𝑢    )
24:22,23: (   𝑢 = 𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑢)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
25:24: (   𝑢 = 𝑣   ▶   ((𝑥 = 𝑢𝑦 = 𝑢) → ( 𝑥 = 𝑢𝑦 = 𝑣))   )
26:25: (   𝑢 = 𝑣   ▶   𝑦((𝑥 = 𝑢𝑦 = 𝑢) → (𝑥 = 𝑢𝑦 = 𝑣))   )
27:26: (   𝑢 = 𝑣   ▶   (∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
28:27: (   𝑢 = 𝑣   ▶   𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
29:28: (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))   )
30:29: (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑢 ) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
31:18,30: (   𝑥𝑥 = 𝑦   ▶   (𝑢 = 𝑣 → ∃𝑥𝑦 (𝑥 = 𝑢𝑦 = 𝑣))   )
qed:31: (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦( 𝑥 = 𝑢𝑦 = 𝑣)))
(∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
 
Theoremax6e2ndVD 43669* The following User's Proof is a Virtual Deduction proof (see wvd1 43330) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2nd 43319 is ax6e2ndVD 43669 without virtual deductions and was automatically derived from ax6e2ndVD 43669. (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 = 𝑢𝑦 = 𝑣))
7:6: (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦 (∃𝑥𝑥 = 𝑢𝑦 = 𝑣))
8:4,7: 𝑦(∃𝑥𝑥 = 𝑢𝑦 = 𝑣)
9:: (𝑧 = 𝑣 → ∀𝑥𝑧 = 𝑣)
10:: (𝑦 = 𝑣 → ∀𝑧𝑦 = 𝑣)
11:: (   𝑧 = 𝑦   ▶   𝑧 = 𝑦   )
12:11: (   𝑧 = 𝑦   ▶   (𝑧 = 𝑣𝑦 = 𝑣)   )
120:11: (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 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 = 𝑢𝑦 = 𝑣))   )
22:21: (¬ ∀𝑥𝑥 = 𝑦 → ((∃𝑥𝑥 = 𝑢 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
23:22: (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
24:: (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦 )
25:23,24: (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
26:14,25: (   ¬ ∀𝑥𝑥 = 𝑦   ▶   𝑦((∃𝑥𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣))   )
27:26: (   ¬ ∀𝑥𝑥 = 𝑦   ▶   (∃𝑦(∃𝑥𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))   )
28:8,27: (   ¬ ∀𝑥𝑥 = 𝑦   ▶   𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣)   )
29:28: (   ¬ ∀𝑥𝑥 = 𝑦   ▶   𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)   )
qed:29: (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑣))
(¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 
Theoremax6e2ndeqVD 43670* The following User's Proof is a Virtual Deduction proof (see wvd1 43330) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2eq 43318 is ax6e2ndeqVD 43670 without virtual deductions and was automatically derived from ax6e2ndeqVD 43670. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (   𝑢𝑣   ▶   𝑢𝑣   )
2:: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   ( 𝑥 = 𝑢𝑦 = 𝑣)   )
3:2: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑥 = 𝑢   )
4:1,3: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑥 𝑣   )
5:2: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑦 = 𝑣   )
6:4,5: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶   𝑥 𝑦   )
7:: (∀𝑥𝑥 = 𝑦𝑥 = 𝑦)
8:7: 𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
9:: 𝑥 = 𝑦𝑥𝑦)
10:8,9: (𝑥𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
11:6,10: (   𝑢𝑣   ,   (𝑥 = 𝑢𝑦 = 𝑣)   ▶    ¬ ∀𝑥𝑥 = 𝑦   )
12:11: (   𝑢𝑣   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
13:12: (   𝑢𝑣   ▶   𝑥((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
14:13: (   𝑢𝑣   ▶   (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥¬ ∀𝑥𝑥 = 𝑦)   )
15:: (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦 )
19:15: (∃𝑥¬ ∀𝑥𝑥 = 𝑦 ↔ ¬ ∀𝑥𝑥 = 𝑦)
20:14,19: (   𝑢𝑣   ▶   (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
21:20: (   𝑢𝑣   ▶   𝑦(∃𝑥(𝑥 = 𝑢 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
22:21: (   𝑢𝑣   ▶   (∃𝑦𝑥(𝑥 = 𝑢 𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦)   )
23:: (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃ 𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
24:22,23: (   𝑢𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦)   )
25:: (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦 )
26:25: (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∃𝑦𝑦¬ 𝑥𝑥 = 𝑦)
260:: (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦𝑦¬ 𝑥𝑥 = 𝑦)
27:260: (∃𝑦𝑦¬ ∀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: (   𝑢𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
31:30: (   𝑢𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣))   )
32:31: (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣)))
33:: (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   )
34:33: (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → 𝑢 = 𝑣)   )
35:34: (   𝑢 = 𝑣   ▶   (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣))   )
36:35: (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣)))
37:: (𝑢 = 𝑣𝑢𝑣)
38:32,36,37: (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ( ¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣))
39:: (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦 (𝑥 = 𝑢𝑦 = 𝑣)))
40:: (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢 𝑦 = 𝑣))
41:40: (¬ ∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥 𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
42:: (∀𝑥𝑥 = 𝑦 ∨ ¬ ∀𝑥𝑥 = 𝑦)
43:39,41,42: (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣 ))
44:40,43: ((¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣) → ∃𝑥 𝑦(𝑥 = 𝑢𝑦 = 𝑣))
qed:38,44: ((¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥 𝑦(𝑥 = 𝑢𝑦 = 𝑣))
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 
Theorem2sb5ndVD 43671* The following User's Proof is a Virtual Deduction proof (see wvd1 43330) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. 2sb5nd 43321 is 2sb5ndVD 43671 without virtual deductions and was automatically derived from 2sb5ndVD 43671. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
2:1: (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑢𝑦 = 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 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
21:2,20: (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
22:21: (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
23:13: (∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
24:22,23: ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
240:24: ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ( 𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
241:: ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ( 𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
242:241,240: ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
243:: ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ( [𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))) ↔ ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))))
25:242,243: (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ([ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
26:: ((¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥 𝑦(𝑥 = 𝑢𝑦 = 𝑣))
qed:25,26: ((¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
 
Theorem2uasbanhVD 43672* The following User's Proof is a Virtual Deduction proof (see wvd1 43330) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. 2uasbanh 43322 is 2uasbanhVD 43672 without virtual deductions and was automatically derived from 2uasbanhVD 43672. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
h1:: (𝜒 ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
100:1: (𝜒 → (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
2:100: (   𝜒   ▶   (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))   )
3:2: (   𝜒   ▶   𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
4:3: (   𝜒   ▶   𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣 )   )
5:4: (   𝜒   ▶   (¬ ∀𝑥𝑥 = 𝑦𝑢 = 𝑣)    )
6:5: (   𝜒   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))   )
7:3,6: (   𝜒   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
8:2: (   𝜒   ▶   𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)   )
9:5: (   𝜒   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜓 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))   )
10:8,9: (   𝜒   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜓   )
101:: ([𝑣 / 𝑦](𝜑𝜓) ↔ ([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
102:101: ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑𝜓) ↔ [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](𝜑𝜓) ↔ ([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) ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)))   )
12:11,110: (   𝜒   ▶   𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓))   )
120:12: (𝜒 → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣 ) ∧ (𝜑𝜓)))
13:1,120: ((∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)) → 𝑥𝑦((𝑥 = 𝑢 𝑦 = 𝑣) ∧ (𝜑𝜓)))
14:: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓 ))   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓))   )
15:14: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓 ))   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
16:14: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓 ))   ▶   (𝜑𝜓)   )
17:16: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓 ))   ▶   𝜑   )
18:15,17: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓 ))   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
19:18: (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓 )) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
20:19: (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑 𝜓)) → ∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
21:20: (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ ( 𝜑𝜓)) → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
22:16: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓 ))   ▶   𝜓   )
23:15,22: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓 ))   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)   )
24:23: (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓 )) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))
25:24: (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑 𝜓)) → ∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))
26:25: (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ ( 𝜑𝜓)) → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))
27:21,26: (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ ( 𝜑𝜓)) → (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ 𝑥𝑦( (𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
qed:13,27: (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ ( 𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ 𝑥𝑦( (𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
(𝜒 ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))       (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
 
Theoreme2ebindVD 43673 The following User's Proof is a Virtual Deduction proof (see wvd1 43330) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. e2ebind 43324 is e2ebindVD 43673 without virtual deductions and was automatically derived from e2ebindVD 43673.
1:: (𝜑𝜑)
2:1: (∀𝑦𝑦 = 𝑥 → (𝜑𝜑))
3:2: (∀𝑦𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑 ))
4:: (   𝑦𝑦 = 𝑥   ▶   𝑦𝑦 = 𝑥   )
5:3,4: (   𝑦𝑦 = 𝑥   ▶   (∃𝑦𝜑 ↔ ∃𝑥 𝜑)   )
6:: (∀𝑦𝑦 = 𝑥 → ∀𝑦𝑦𝑦 = 𝑥)
7:5,6: (   𝑦𝑦 = 𝑥   ▶   𝑦(∃𝑦𝜑 𝑥𝜑)   )
8:7: (   𝑦𝑦 = 𝑥   ▶   (∃𝑦𝑦𝜑 𝑦𝑥𝜑)   )
9:: (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
10:8,9: (   𝑦𝑦 = 𝑥   ▶   (∃𝑦𝑦𝜑 𝑥𝑦𝜑)   )
11:: (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
12:11: (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
13:10,12: (   𝑦𝑦 = 𝑥   ▶   (∃𝑥𝑦𝜑 𝑦𝜑)   )
14:13: (∀𝑦𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃ 𝑦𝜑))
15:: (∀𝑦𝑦 = 𝑥 ↔ ∀𝑥𝑥 = 𝑦)
qed:14,15: (∀𝑥𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃ 𝑦𝜑))
(Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
 
21.39.8  Virtual Deduction transcriptions of textbook proofs
 
Theoremsb5ALTVD 43674* 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 2268, 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 43286 is sb5ALTVD 43674 without virtual deductions and was automatically derived from sb5ALTVD 43674.
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 = 𝑦𝜑 )   )
5:4: ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑) )
6:: (   𝑥(𝑥 = 𝑦𝜑)   ▶   𝑥(𝑥 = 𝑦𝜑)   )
7:: (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑 )   ▶   (𝑥 = 𝑦𝜑)   )
8:7: (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑 )   ▶   𝜑   )
9:7: (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑 )   ▶   𝑥 = 𝑦   )
10:8,9: (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑 )   ▶   [𝑦 / 𝑥]𝜑   )
101:: ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
11:101,10: (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑 )
12:5,11: (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑 )) ∧ (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
qed:12: ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑) )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremvk15.4jVD 43675 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 43289 is vk15.4jVD 43675 without virtual deductions and was automatically derived from vk15.4jVD 43675. 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(𝜃𝜏 ))
h3:: ¬ ∀𝑥(𝜏𝜑)
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)   ▶   (𝜃𝜏)   )
12:11: (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝑥(𝜃𝜏)   )
13:12: (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ ¬ ∃𝑥(𝜃𝜏)   )
14:2,13: (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ ∀𝑥𝜒   )
140:: (∃𝑥¬ 𝜃 → ∀𝑥𝑥¬ 𝜃 )
141:140: (¬ ∃𝑥¬ 𝜃 → ∀𝑥¬ ∃𝑥 ¬ 𝜃)
142:: (∀𝑥𝜒 → ∀𝑥𝑥𝜒)
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 → ∀𝑥𝑥¬ 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   ▶   (𝜓𝜒 )   )
23:21,22: (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶   ¬ 𝜓   )
24:23: (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶    𝑥¬ 𝜓   )
240:: (∃𝑥¬ 𝜓 → ∀𝑥𝑥¬ 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(𝜃𝜏))    &    ¬ ∀𝑥(𝜏𝜑)       (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)
 
TheoremnotnotrALTVD 43676 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 43290 is notnotrALTVD 43676 without virtual deductions and was automatically derived from notnotrALTVD 43676. 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   ▶   (¬ ¬ 𝜑𝜑)   )
6:5,1: (   ¬ ¬ 𝜑   ▶   𝜑   )
qed:6: (¬ ¬ 𝜑𝜑)
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)
 
Theoremcon3ALTVD 43677 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 43291 is con3ALTVD 43677 without virtual deductions and was automatically derived from con3ALTVD 43677. Step i of the User's Proof corresponds to step i of the Fitch-style proof.
1:: (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2:: (   (𝜑𝜓)   ,   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜑   )
3:: (¬ ¬ 𝜑𝜑)
4:2: (   (𝜑𝜓)   ,   ¬ ¬ 𝜑   ▶   𝜑   )
5:1,4: (   (𝜑𝜓)   ,   ¬ ¬ 𝜑   ▶   𝜓   )
6:: (𝜓 → ¬ ¬ 𝜓)
7:6,5: (   (𝜑𝜓)   ,   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜓   )
8:7: (   (𝜑𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓 )   )
9:: ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 ¬ 𝜑))
10:8: (   (𝜑𝜓)   ▶   𝜓 → ¬ 𝜑)   )
qed:10: ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
21.39.9  Theorems proved using conjunction-form Virtual Deduction
 
TheoremelpwgdedVD 43678 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 43325 is elpwgdedVD 43678 using conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝐴 ∈ V   )    &   (   𝜓   ▶   𝐴𝐵   )       (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )
 
Theoremsspwimp 43679 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 43679, using conventional notation, was translated from virtual deduction form, sspwimpVD 43680, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
TheoremsspwimpVD 43680 The following User's Proof is a Virtual Deduction proof (see wvd1 43330) 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 43679 is sspwimpVD 43680 without virtual deductions and was derived from sspwimpVD 43680. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (   .............. 𝑥 ∈ 𝒫 𝐴    ▶   𝑥 ∈ 𝒫 𝐴   )
3:2: (   .............. 𝑥 ∈ 𝒫 𝐴    ▶   𝑥𝐴   )
4:3,1: (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥𝐵   )
5:: 𝑥 ∈ V
6:4,5: (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵    )
7:6: (   𝐴𝐵   ▶   (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)    )
8:7: (   𝐴𝐵   ▶   𝑥(𝑥 ∈ 𝒫 𝐴𝑥 𝒫 𝐵)   )
9:8: (   𝐴𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
qed:9: (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
Theoremsspwimpcf 43681 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 43681, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 43682, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
TheoremsspwimpcfVD 43682 The following User's Proof is a Virtual Deduction proof (see wvd1 43330) 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 43681 is sspwimpcfVD 43682 without virtual deductions and was derived from sspwimpcfVD 43682. 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:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (   ........... 𝑥 ∈ 𝒫 𝐴    ▶   𝑥 ∈ 𝒫 𝐴   )
3:2: (   ........... 𝑥 ∈ 𝒫 𝐴    ▶   𝑥𝐴   )
4:3,1: (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥𝐵   )
5:: 𝑥 ∈ V
6:4,5: (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵    )
7:6: (   𝐴𝐵   ▶   (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)    )
8:7: (   𝐴𝐵   ▶   𝑥(𝑥 ∈ 𝒫 𝐴𝑥 𝒫 𝐵)   )
9:8: (   𝐴𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
qed:9: (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
TheoremsuctrALTcf 43683 The sucessor of a transitive class is transitive. suctrALTcf 43683, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 43684, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)
 
TheoremsuctrALTcfVD 43684 The following User's Proof is a Virtual Deduction proof (see wvd1 43330) 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 43683 is suctrALTcfVD 43684 without virtual deductions and was derived automatically from suctrALTcfVD 43684. 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:: (   ......... (𝑧𝑦𝑦 suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
3:2: (   ......... (𝑧𝑦𝑦 suc 𝐴)   ▶   𝑧𝑦   )
4:: (   ................................... ....... 𝑦𝐴   ▶   𝑦𝐴   )
5:1,3,4: (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴) , 𝑦𝐴   )   ▶   𝑧𝐴   )
6:: 𝐴 ⊆ suc 𝐴
7:5,6: (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴) , 𝑦𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
8:7: (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)    )   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
9:: (   ................................... ...... 𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
10:3,9: (   ........ (   (𝑧𝑦𝑦 suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧𝐴   )
11:10,6: (   ........ (   (𝑧𝑦𝑦 suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
12:11: (   .......... (𝑧𝑦𝑦 suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
13:2: (   .......... (𝑧𝑦𝑦 suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
14:13: (   .......... (𝑧𝑦𝑦 suc 𝐴)   ▶   (𝑦𝐴𝑦 = 𝐴)   )
15:8,12,14: (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)    )   ▶   𝑧 ∈ suc 𝐴   )
16:15: (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
17:16: (   Tr 𝐴   ▶   𝑧𝑦((𝑧 𝑦𝑦 ∈ 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
 
TheoremsuctrALT3 43685 The successor of a transitive class is transitive. suctrALT3 43685 is the completed proof in conventional notation of the Virtual Deduction proof https://us.metamath.org/other/completeusersproof/suctralt3vd.html 43685. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 43330 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 43327). 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)
 
TheoremsspwimpALT 43686 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 43686 is the completed proof in conventional notation of the Virtual Deduction proof https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html 43686. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 43330 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 43325). 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.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
TheoremunisnALT 43687 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 43687 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 43687. 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 43687, 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.

 
TheoremnotnotrALT2 43688 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.)
(¬ ¬ 𝜑𝜑)
 
TheoremsspwimpALT2 43689 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.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
Theoreme2ebindALT 43690 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 43673. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
 
Theoremax6e2ndALT 43691* If at least two sets exist (dtru 5437), then the same is true expressed in an alternate form similar to the form of ax6e 2383. 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 43669. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 
Theoremax6e2ndeqALT 43692* "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 2383 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 43670. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 
Theorem2sb5ndALT 43693* Equivalence for double substitution 2sb5 2272 without distinct 𝑥, 𝑦 requirement. 2sb5nd 43321 is derived from 2sb5ndVD 43671. 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 43671. (Contributed by Alan Sare, 19-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
 
TheoremchordthmALT 43694* 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 26341 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 26342. https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 26342 is a Virtual Deduction User's Proof transcription of chordthm 26342. That VD User's Proof was input into completeusersproof, automatically generating this chordthmALT 43694 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))))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝑃 ∈ ℂ)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → ((𝐴𝑃)𝐹(𝐵𝑃)) = π)    &   (𝜑 → ((𝐶𝑃)𝐹(𝐷𝑃)) = π)    &   (𝜑𝑄 ∈ ℂ)    &   (𝜑 → (abs‘(𝐴𝑄)) = (abs‘(𝐵𝑄)))    &   (𝜑 → (abs‘(𝐴𝑄)) = (abs‘(𝐶𝑄)))    &   (𝜑 → (abs‘(𝐴𝑄)) = (abs‘(𝐷𝑄)))       (𝜑 → ((abs‘(𝑃𝐴)) · (abs‘(𝑃𝐵))) = ((abs‘(𝑃𝐶)) · (abs‘(𝑃𝐷))))
 
Theoremisosctrlem1ALT 43695 Lemma for isosctr 26326. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 26326. As it is verified by the Metamath program, isosctrlem1ALT 43695 verifies https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 43695. (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)
 
Theoremiunconnlem2 43696* 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 43696 verifies https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 43696. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 ↔ ((((((𝜑𝑢𝐽) ∧ 𝑣𝐽) ∧ (𝑢 𝑘𝐴 𝐵) ≠ ∅) ∧ (𝑣 𝑘𝐴 𝐵) ≠ ∅) ∧ (𝑢𝑣) ⊆ (𝑋 𝑘𝐴 𝐵)) ∧ 𝑘𝐴 𝐵 ⊆ (𝑢𝑣)))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝐵𝑋)    &   ((𝜑𝑘𝐴) → 𝑃𝐵)    &   ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Conn)       (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)
 
TheoremiunconnALT 43697* 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 43697 verifies https://us.metamath.org/other/completeusersproof/iunconaltvd.html 43697. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝐵𝑋)    &   ((𝜑𝑘𝐴) → 𝑃𝐵)    &   ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Conn)       (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)
 
Theoremsineq0ALT 43698 A complex number whose sine is zero is an integer multiple of π. The Virtual Deduction form of the proof is https://us.metamath.org/other/completeusersproof/sineq0altvd.html. The Metamath form of the proof is sineq0ALT 43698. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 26033. The Virtual Deduction proof is verified by automatically transforming it into the Metamath form of the proof using completeusersproof, which is verified by the Metamath program. The proof of https://us.metamath.org/other/completeusersproof/sineq0altro.html 26033 is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ℂ → ((sin‘𝐴) = 0 ↔ (𝐴 / π) ∈ ℤ))
 
21.40  Mathbox for Glauco Siliprandi
 
21.40.1  Miscellanea
 
Theoremevth2f 43699* A version of evth2 24476 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑦𝐹    &   𝑥𝑋    &   𝑦𝑋    &   𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑥) ≤ (𝐹𝑦))
 
Theoremelunif 43700* A version of eluni 4912 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
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