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Theorem List for Metamath Proof Explorer - 40001-40100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3orbi123VD 40001 Virtual deduction proof of 3orbi123 39653. 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 39778 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   (𝜑𝜓)   )
3:1,?: e1a 39778 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   (𝜒𝜃)   )
4:1,?: e1a 39778 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   (𝜏𝜂)   )
5:2,3,?: e11 39839 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   ((𝜑𝜒) ↔ (𝜓𝜃))   )
6:5,4,?: e11 39839 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   (((𝜑𝜒) ∨ 𝜏) ↔ ((𝜓𝜃) 𝜂))   )
7:?: (((𝜑𝜒) ∨ 𝜏) ↔ (𝜑 𝜒𝜏))
8:6,7,?: e10 39845 (   ((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂))   ▶   ((𝜑𝜒𝜏) ↔ ((𝜓𝜃) 𝜂))   )
9:?: (((𝜓𝜃) ∨ 𝜂) ↔ (𝜓𝜃𝜂))
10:8,9,?: e10 39845 (   ((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 40002 Virtual deduction proof of the analogue of sbcor 3696 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 39778 (   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 39845 (   𝐴𝐵   ▶   [𝐴 / 𝑥](((𝜑 𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))   )
4:1,33,?: e11 39839 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))   )
5:2,4,?: e11 39839 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
6:1,?: e1a 39778 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
7:6,?: e1a 39778 (   𝐴𝐵   ▶   (([𝐴 / 𝑥](𝜑 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) [𝐴 / 𝑥]𝜒))   )
8:5,7,?: e11 39839 (   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 39845 (   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 40003* Virtual deduction proof of alrim3con13v 39675. 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 39782 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝜓   )
4:2,?: e2 39782 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝜑   )
5:2,?: e2 39782 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝜒   )
6:1,4,?: e12 39875 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝑥𝜑   )
7:3,?: e2 39782 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝑥𝜓   )
8:5,?: e2 39782 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   𝑥𝜒   )
9:7,6,8,?: e222 39787 (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 𝜑𝜒)   ▶   (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒)   )
10:9,?: e2 39782 (   (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 40004 Virtual deduction proof of exbir 39620. 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 39875 (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓)   ▶   (𝜒𝜃)   )
6:3,5,?: e32 39909 (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓), 𝜃   ▶   𝜒   )
7:6: (   ((𝜑𝜓) → (𝜒 𝜃)), (𝜑𝜓)   ▶   (𝜃𝜒)   )
8:7: (   ((𝜑𝜓) → (𝜒𝜃))    ▶   ((𝜑𝜓) → (𝜃𝜒))   )
9:8,?: e1a 39778 (   ((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 40005 Virtual deduction proof of exbiri 801. 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 39845 (   𝜑   ▶   (𝜓 → (𝜒𝜃))   )
6:3,5,?: e21 39881 (   𝜑   ,   𝜓   ▶   (𝜒𝜃)   )
7:4,6,?: e32 39909 (   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 40006* Virtual deduction proof of rspsbc2 39676. 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 39899 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   [𝐴 / 𝑥]𝑦𝐷𝜑   )
5:1,4,?: e13 39899 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑦𝐷[𝐴 / 𝑥]𝜑   )
6:2,5,?: e23 39906 (   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 40007 Virtual deduction proof of 3impexp 1420. 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 39845 (   ((𝜑𝜓𝜒) 𝜃)   ▶   (((𝜑𝜓) ∧ 𝜒) → 𝜃)   )
4:3,?: e1a 39778 (   ((𝜑𝜓𝜒) 𝜃)   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
5:4,?: e1a 39778 (   ((𝜑𝜓𝜒) 𝜃)   ▶   (𝜑 → (𝜓 → (𝜒𝜃)))   )
6:5: (((𝜑𝜓𝜒) → 𝜃) → (𝜑 → (𝜓 → (𝜒𝜃))))
7:: (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   (𝜑 → (𝜓 → (𝜒𝜃)))   )
8:7,?: e1a 39778 (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
9:8,?: e1a 39778 (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   (((𝜑𝜓) ∧ 𝜒) → 𝜃)   )
10:2,9,?: e01 39842 (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   ((𝜑𝜓𝜒) → 𝜃)   )
11:10: ((𝜑 → (𝜓 → (𝜒 𝜃))) → ((𝜑𝜓𝜒) → 𝜃))
qed:6,11,?: e00 39919 (((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 40008 Virtual deduction proof of 3impexpbicom 39621. 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 39845 (   ((𝜑𝜓𝜒) → (𝜃𝜏))   ▶   ((𝜑𝜓𝜒) → (𝜏𝜃))   )
4:3,?: e1a 39778 (   ((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 39778 (   (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))   ▶   ((𝜑𝜓𝜒) → (𝜏 𝜃))   )
8:7,2,?: e10 39845 (   (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))   ▶   ((𝜑𝜓𝜒) → (𝜃 𝜏))   )
9:8: ((𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))) → ((𝜑𝜓𝜒) → (𝜃 𝜏)))
qed:5,9,?: e00 39919 (((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 40009 Virtual deduction proof of 3impexpbicomi 39622. 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 39923 (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 40010* Virtual deduction proof of sbcoreleleq 39677. 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 39778 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 𝑦𝑥𝐴)   )
3:1,?: e1a 39778 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑦 𝑥𝐴𝑥)   )
4:1,?: e1a 39778 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)   )
5:2,3,4,?: e111 39825 (   𝐴𝐵   ▶   ((𝑥𝐴 𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥 [𝐴 / 𝑦]𝑥 = 𝑦))   )
6:1,?: e1a 39778 (   𝐴𝐵    ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥 𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
7:5,6: e11 39839 (   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 40011* Virtual deduction proof of nfra2 3127. 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 39919 (∀𝑥𝐴𝑦𝐵𝜑 𝑦𝑦𝐵𝑥𝐴𝜑)
4:2: 𝑦(∀𝑥𝐴𝑦𝐵𝜑 𝑦𝐵𝑥𝐴𝜑)
5:4,?: e0a 39923 (∀𝑦𝑥𝐴𝑦𝐵𝜑 𝑦𝑦𝐵𝑥𝐴𝜑)
qed:3,5,?: e00 39919 (∀𝑥𝐴𝑦𝐵𝜑 𝑦𝑥𝐴𝑦𝐵𝜑)
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝑥𝐴𝑦𝐵 𝜑)
 
TheoremtratrbVD 40012* Virtual deduction proof of tratrb 39678. 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 39778 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐴   )
3:1,?: e1a 39778 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)    ▶   𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
4:1,?: e1a 39778 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝐵𝐴   )
5:: (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥𝑦𝑦𝐵)   )
6:5,?: e2 39782 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑥𝑦   )
7:5,?: e2 39782 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑦𝐵   )
8:2,7,4,?: e121 39807 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   𝑦𝐴   )
9:2,6,8,?: e122 39804 (   (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 39786 (   (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 39878 (   (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 39906 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵), 𝑥 = 𝐵   ▶   𝑦𝑥   )
17:6,16,?: e23 39906 (   (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 39878 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   ¬ 𝑥 = 𝐵   )
21:3,?: e1a 39778 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦𝐴 𝑥𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
22:21,9,4,?: e121 39807 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥 𝑥 = 𝑦)   )
23:22,?: e2 39782 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
24:4,23,?: e12 39875 (   (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴), (𝑥𝑦 𝑦𝐵)   ▶   (𝑥𝐵𝐵𝑥𝑥 = 𝐵)   )
25:14,20,24,?: e222 39787 (   (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 39923 ((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 39923 ((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 39778 (   (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 40013 Virtual deduction proof of al2im 1858. 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 39778 (   𝑥(𝜑 → (𝜓𝜒))    ▶   (∀𝑥𝜑 → ∀𝑥(𝜓𝜒))   )
3:: (∀𝑥(𝜓𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒))
4:2,3,?: e10 39845 (   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 40014 Virtual deduction proof of syl5imp 39654. 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 39778 (   (𝜑 → (𝜓𝜒))   ▶   (𝜓 → (𝜑𝜒))   )
3:: (   (𝜑 → (𝜓𝜒))   ,   (𝜃 𝜓)   ▶   (𝜃𝜓)   )
4:3,2,?: e21 39881 (   (𝜑 → (𝜓𝜒))   ,   (𝜃 𝜓)   ▶   (𝜃 → (𝜑𝜒))   )
5:4,?: e2 39782 (   (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 40015 Virtual deduction proof of idiALT 39619. 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 39923 𝜑
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑       𝜑
 
TheoremancomstVD 40016 Closed form of ancoms 452. 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 39923 (((𝜑𝜓) → 𝜒) ↔ ((𝜓 𝜑) → 𝜒))
The proof of ancomst 458 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 40017* Quantification restricted to a subclass for two quantifiers. ssralv 3884 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 39673 is ssralv2VD 40017 without virtual deductions and was automatically derived from ssralv2VD 40017.
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 40018 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5998 using the Axiom of Regularity indirectly through dford2 8814. 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 39679 is ordelordALTVD 40018 without virtual deductions and was automatically derived from ordelordALTVD 40018 using the tools program translate..without..overwriting.cmd and the Metamath program "MM-PA> MINIMIZEWITH *" 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 40019 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 3980 is equncomVD 40019 without virtual deductions and was automatically derived from equncomVD 40019.
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 40020 Inference form of equncom 3980. 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 3981 is equncomiVD 40020 without virtual deductions and was automatically derived from equncomiVD 40020.
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 40021 A set belongs to its successor. Alternate proof of sucid 6055. 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 40022 is sucidALTVD 40021 without virtual deductions and was automatically derived from sucidALTVD 40021. 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 5982, 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 8814.
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 40022 A set belongs to its successor. This proof was automatically derived from sucidALTVD 40021 using translatewithout_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 40023 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 6055 is sucidVD 40023 without virtual deductions and was automatically derived from sucidVD 40023.
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 40024 Implication form of imbi12i 342. 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 338 is imbi12VD 40024 without virtual deductions and was automatically derived from imbi12VD 40024.
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 40025 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 39662 is imbi13VD 40025 without virtual deductions and was automatically derived from imbi13VD 40025.
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 40026 Distribution of class substitution over a left-nested implication. Similar to sbcimg 3694. 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 39680 is sbcim2gVD 40026 without virtual deductions and was automatically derived from sbcim2gVD 40026.
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 40027 Implication form of sbcbii 3702. 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 39681 is sbcbiVD 40027 without virtual deductions and was automatically derived from sbcbiVD 40027.
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 40028* 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 39682 is trsbcVD 40028 without virtual deductions and was automatically derived from trsbcVD 40028.
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 40029* 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 39683 is truniALTVD 40029 without virtual deductions and was automatically derived from truniALTVD 40029.
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 40030 Non-virtual deduction form of e33 39885. 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 39663 is ee33VD 40030 without virtual deductions and was automatically derived from ee33VD 40030.
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 40031* The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 40032. 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 40032 is trintALTVD 40031 without virtual deductions and was automatically derived from trintALTVD 40031.
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 40032* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 40032 is an alternate proof of trint 5003. trintALT 40032 is trintALTVD 40031 without virtual deductions and was automatically derived from trintALTVD 40031 using the tools program translate..without..overwriting.cmd and the Metamath program "MM-PA> MINIMIZEWITH *" command. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 
Theoremundif3VD 40033 The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 4114. 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 4114 is undif3VD 40033 without virtual deductions and was automatically derived from undif3VD 40033.
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 40034 Virtual deduction proof of sbcssg 4305. 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 4305 is sbcssgVD 40034 without virtual deductions and was automatically derived from sbcssgVD 40034.
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 40035 Virtual deduction proof of csbin 4235. 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 4235 is csbingVD 40035 without virtual deductions and was automatically derived from csbingVD 40035.
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 40036* Virtual deduction proof of onfrALTlem5 39684. 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 39684 is onfrALTlem5VD 40036 without virtual deductions and was automatically derived from onfrALTlem5VD 40036.
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 40037* Virtual deduction proof of onfrALTlem4 39685. 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 39685 is onfrALTlem4VD 40037 without virtual deductions and was automatically derived from onfrALTlem4VD 40037.
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 40038* Virtual deduction proof of onfrALTlem3 39686. 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 39686 is onfrALTlem3VD 40038 without virtual deductions and was automatically derived from onfrALTlem3VD 40038.
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 40039 Virtual deduction proof of simplbi2comt 497. 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 497 is simplbi2comtVD 40039 without virtual deductions and was automatically derived from simplbi2comtVD 40039.
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 40040* Virtual deduction proof of onfrALTlem2 39688. 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 39688 is onfrALTlem2VD 40040 without virtual deductions and was automatically derived from onfrALTlem2VD 40040.
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 40041* Virtual deduction proof of onfrALTlem1 39690. 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 39690 is onfrALTlem1VD 40041 without virtual deductions and was automatically derived from onfrALTlem1VD 40041.
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 40042 Virtual deduction proof of onfrALT 39691. 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 39691 is onfrALTVD 40042 without virtual deductions and was automatically derived from onfrALTVD 40042.
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 40043 Virtual deduction proof of csbeq2 3754. 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 3754 is csbeq2gVD 40043 without virtual deductions and was automatically derived from csbeq2gVD 40043.
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 40044 Virtual deduction proof of csbsng 4474. 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 4474 is csbsngVD 40044 without virtual deductions and was automatically derived from csbsngVD 40044.
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 40045 Virtual deduction proof of csbxp 5448. 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 5448 is csbxpgVD 40045 without virtual deductions and was automatically derived from csbxpgVD 40045.
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 40046 Virtual deduction proof of csbres 5645. 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 5645 is csbresgVD 40046 without virtual deductions and was automatically derived from csbresgVD 40046.
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 40047 Virtual deduction proof of csbrn 5850. 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 5850 is csbrngVD 40047 without virtual deductions and was automatically derived from csbrngVD 40047.
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 40048 Virtual deduction proof of csbima12 5737. 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 5737 is csbima12gALTVD 40048 without virtual deductions and was automatically derived from csbima12gALTVD 40048.
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 40049 Virtual deduction proof of csbuni 4701. 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 4701 is csbunigVD 40049 without virtual deductions and was automatically derived from csbunigVD 40049.
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 40050 Virtual deduction proof of csbfv12 6490. 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 6490 is csbfv12gALTVD 40050 without virtual deductions and was automatically derived from csbfv12gALTVD 40050.
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 40051 Virtual deduction proof of con5 39664. 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 39664 is con5VD 40051 without virtual deductions and was automatically derived from con5VD 40051.
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 40052 Virtual deduction proof of relopab 5493. 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 5493 is relopabVD 40052 without virtual deductions and was automatically derived from relopabVD 40052.
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 40053 Virtual deduction proof of 19.41rg 39692. 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 39692 is 19.41rgVD 40053 without virtual deductions and was automatically derived from 19.41rgVD 40053. (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 40054 Virtual deduction proof of 2pm13.193 39694. 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 39694 is 2pm13.193VD 40054 without virtual deductions and was automatically derived from 2pm13.193VD 40054. (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 40055 Virtual deduction proof of hbimpg 39696. 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 39696 is hbimpgVD 40055 without virtual deductions and was automatically derived from hbimpgVD 40055. (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 40056 Virtual deduction proof of hbalg 39697. 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 39697 is hbalgVD 40056 without virtual deductions and was automatically derived from hbalgVD 40056. (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 40057 Virtual deduction proof of hbexg 39698. 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 39698 is hbexgVD 40057 without virtual deductions and was automatically derived from hbexgVD 40057. (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 40058* The following User's Proof is a Virtual Deduction proof (see wvd1 39711) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2eq 39699 is ax6e2eqVD 40058 without virtual deductions and was automatically derived from ax6e2eqVD 40058. (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 40059* The following User's Proof is a Virtual Deduction proof (see wvd1 39711) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2nd 39700 is ax6e2ndVD 40059 without virtual deductions and was automatically derived from ax6e2ndVD 40059. (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 40060* The following User's Proof is a Virtual Deduction proof (see wvd1 39711) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2eq 39699 is ax6e2ndeqVD 40060 without virtual deductions and was automatically derived from ax6e2ndeqVD 40060. (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 40061* The following User's Proof is a Virtual Deduction proof (see wvd1 39711) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. 2sb5nd 39702 is 2sb5ndVD 40061 without virtual deductions and was automatically derived from 2sb5ndVD 40061. (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((