P. 401 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

3orbi123VD 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::	⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   )
2:1,?: e1a 39778	⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (𝜑 ↔ 𝜓)   )
3:1,?: e1a 39778	⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (𝜒 ↔ 𝜃)   )
4:1,?: e1a 39778	⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (𝜏 ↔ 𝜂)   )
5:2,3,?: e11 39839	⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃))   )
6:5,4,?: e11 39839	⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))   )
7:?:	⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ (𝜑 ∨ 𝜒 ∨ 𝜏))
8:6,7,?: e10 39845	⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))   )
9:?:	⊢ (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))
10:8,9,?: e10 39845	⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))   )
qed:10:	⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)))

Theorem

sbc3orgVD 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::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2:1,?: e1a 39778	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
3::	⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒))
32:3:	⊢ ∀𝑥(((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒))
33:1,32,?: e10 39845	⊢ (   𝐴 ∈ 𝐵   ▶   [𝐴 / 𝑥](((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒))   )
4:1,33,?: e11 39839	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒))   )
5:2,4,?: e11 39839	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
6:1,?: e1a 39778	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))   )
7:6,?: e1a 39778	⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
8:5,7,?: e11 39839	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
9:?:	⊢ ((([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))
10:8,9,?: e10 39845	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))   )
qed:10:	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒)))

Theorem

19.21a3con13vVD 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)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   (𝜓 ∧ 𝜑 ∧ 𝜒)   )
3:2,?: e2 39782	⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   𝜓   )
4:2,?: e2 39782	⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   𝜑   )
5:2,?: e2 39782	⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   𝜒   )
6:1,4,?: e12 39875	⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   ∀𝑥𝜑   )
7:3,?: e2 39782	⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   ∀𝑥𝜓   )
8:5,?: e2 39782	⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   ∀𝑥𝜒   )
9:7,6,8,?: e222 39787	⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒)   )
10:9,?: e2 39782	⊢ (   (𝜑 → ∀𝑥𝜑)   ,   (𝜓 ∧ 𝜑 ∧ 𝜒)   ▶   ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)   )
11:10:in2	⊢ (   (𝜑 → ∀𝑥𝜑)   ▶   ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒))   )
qed:11:in1	⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))

Theorem

exbirVD 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::	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ▶   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   )
2::	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,    (𝜑 ∧ 𝜓)   ▶   (𝜑 ∧ 𝜓)   )
3::	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,    (𝜑 ∧ 𝜓), 𝜃   ▶   𝜃   )
5:1,2,?: e12 39875	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓)   ▶   (𝜒 ↔ 𝜃)   )
6:3,5,?: e32 39909	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓), 𝜃   ▶   𝜒   )
7:6:	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓)   ▶   (𝜃 → 𝜒)   )
8:7:	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ▶   ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))   )
9:8,?: e1a 39778	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ▶   (𝜑 → (𝜓 → (𝜃 → 𝜒)))   )
qed:9:	⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))

(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))

Theorem

exbiriVD 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::	⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
2::	⊢ (   𝜑   ▶   𝜑   )
3::	⊢ (   𝜑   ,   𝜓   ▶   𝜓   )
4::	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   )
5:2,1,?: e10 39845	⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   )
6:3,5,?: e21 39881	⊢ (   𝜑   ,   𝜓   ▶   (𝜒 ↔ 𝜃)   )
7:4,6,?: e32 39909	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
8:7:	⊢ (   𝜑   ,   𝜓   ▶   (𝜃 → 𝜒)   )
9:8:	⊢ (   𝜑   ▶   (𝜓 → (𝜃 → 𝜒))   )
qed:9:	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

Theorem

rspsbc2VD 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::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2::	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   𝐶 ∈ 𝐷   )
3::	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑   )
4:1,3,?: e13 39899	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑   )
5:1,4,?: e13 39899	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑   )
6:2,5,?: e23 39906	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   )
7:6:	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   )
8:7:	⊢ (   𝐴 ∈ 𝐵   ▶   (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   )
qed:8:	⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

Theorem

3impexpVD 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::	⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   )
2::	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
3:1,2,?: e10 39845	⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)   )
4:3,?: e1a 39778	⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))   )
5:4,?: e1a 39778	⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   )
6:5:	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) → (𝜑 → (𝜓 → (𝜒 → 𝜃))))
7::	⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   )
8:7,?: e1a 39778	⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))   )
9:8,?: e1a 39778	⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)   )
10:2,9,?: e01 39842	⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   )
11:10:	⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
qed:6,11,?: e00 39919	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))

Theorem

3impexpbicomVD 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::	⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))   )
2::	⊢ ((𝜃 ↔ 𝜏) ↔ (𝜏 ↔ 𝜃))
3:1,2,?: e10 39845	⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃))   )
4:3,?: e1a 39778	⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))   ▶   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   )
5:4:	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) → (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))))
6::	⊢ (   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   ▶   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   )
7:6,?: e1a 39778	⊢ (   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃))   )
8:7,2,?: e10 39845	⊢ (   (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))   )
9:8:	⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))
qed:5,9,?: e00 39919	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))))

Theorem

3impexpbicomiVD 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::	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))
qed:1,?: e0a 39923	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))

Theorem

sbcoreleleqVD 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::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2:1,?: e1a 39778	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴)   )
3:1,?: e1a 39778	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦]𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)   )
4:1,?: e1a 39778	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦 ↔ 𝑥 = 𝐴)   )
5:2,3,4,?: e111 39825	⊢ (   𝐴 ∈ 𝐵   ▶   ((𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥 ∨ [𝐴 / 𝑦]𝑥 = 𝑦))   )
6:1,?: e1a 39778	⊢ (   𝐴 ∈ 𝐵    ▶   ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥 ∨ [𝐴 / 𝑦]𝑥 = 𝑦))   )
7:5,6: e11 39839	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴))   )
qed:7:	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)))

Theorem

hbra2VD 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::	⊢ (∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑 → ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
2::	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
3:1,2,?: e00 39919	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 → ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
4:2:	⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
5:4,?: e0a 39923	⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
qed:3,5,?: e00 39919	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 → ∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑)

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

Theorem

tratrbVD 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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)    ▶   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   )
2:1,?: e1a 39778	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐴   )
3:1,?: e1a 39778	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)    ▶   ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   )
4:1,?: e1a 39778	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ∈ 𝐴   )
5::	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   )
6:5,?: e2 39782	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑥 ∈ 𝑦   )
7:5,?: e2 39782	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑦 ∈ 𝐵   )
8:2,7,4,?: e121 39807	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑦 ∈ 𝐴   )
9:2,6,8,?: e122 39804	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑥 ∈ 𝐴   )
10::	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥   ▶   𝐵 ∈ 𝑥   )
11:6,7,10,?: e223 39786	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥   ▶   (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥)   )
12:11:	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   (𝐵 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥))   )
13::	⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥)
14:12,13,?: e20 39878	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   ¬ 𝐵 ∈ 𝑥   )
15::	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
16:7,15,?: e23 39906	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵   ▶   𝑦 ∈ 𝑥   )
17:6,16,?: e23 39906	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵   ▶   (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)   )
18:17:	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   (𝑥 = 𝐵 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥))   )
19::	⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)
20:18,19,?: e20 39878	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   ¬ 𝑥 = 𝐵   )
21:3,?: e1a 39778	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   )
22:21,9,4,?: e121 39807	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   )
23:22,?: e2 39782	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   [𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   )
24:4,23,?: e12 39875	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   (𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵)   )
25:14,20,24,?: e222 39787	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑥 ∈ 𝐵   )
26:25:	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)   )
27::	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
28:27,?: e0a 39923	⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
29:28,26:	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)    ▶   ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)   )
30::	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑥∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
31:30,?: e0a 39923	⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
32:31,29:	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)   )
33:32,?: e1a 39778	⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐵   )
qed:33:	⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)

Theorem

al2imVD 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 → (𝜓 → 𝜒))    ▶   ∀𝑥(𝜑 → (𝜓 → 𝜒))   )
2:1,?: e1a 39778	⊢ (   ∀𝑥(𝜑 → (𝜓 → 𝜒))    ▶   (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒))   )
3::	⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒))
4:2,3,?: e10 39845	⊢ (   ∀𝑥(𝜑 → (𝜓 → 𝜒))    ▶   (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))   )
qed:4:	⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Theorem

syl5impVD 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 → (𝜓 → 𝜒))   ▶   (𝜑 → (𝜓 → 𝜒))   )
2:1,?: e1a 39778	⊢ (   (𝜑 → (𝜓 → 𝜒))   ▶   (𝜓 → (𝜑 → 𝜒))   )
3::	⊢ (   (𝜑 → (𝜓 → 𝜒))   ,   (𝜃 → 𝜓)   ▶   (𝜃 → 𝜓)   )
4:3,2,?: e21 39881	⊢ (   (𝜑 → (𝜓 → 𝜒))   ,   (𝜃 → 𝜓)   ▶   (𝜃 → (𝜑 → 𝜒))   )
5:4,?: e2 39782	⊢ (   (𝜑 → (𝜓 → 𝜒))   ,   (𝜃 → 𝜓)   ▶   (𝜑 → (𝜃 → 𝜒))   )
6:5:	⊢ (   (𝜑 → (𝜓 → 𝜒))   ▶   ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒)))   )
qed:6:	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒))))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒))))

Theorem

idiVD 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

Theorem

ancomstVD 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::	⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
qed:1,?: e0a 39923	⊢ (((𝜑 ∧ 𝜓) → 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.)

⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒))

Theorem

ssralv2VD 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::	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   )
2::	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑   )
3:1:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   𝐴 ⊆ 𝐵   )
4:3,2:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐷𝜑   )
5:4:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑)   )
6:5:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑)   )
7::	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
8:7,6:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴   ▶   ∀𝑦 ∈ 𝐷𝜑   )
9:1:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ▶   𝐶 ⊆ 𝐷   )
10:9,8:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴   ▶   ∀𝑦 ∈ 𝐶𝜑   )
11:10:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑)   )
12::	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ∀𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷))
13::	⊢ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑)
14:12,13,11:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑)   )
15:14:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑   )
16:15:	⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)    ▶   (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑)   )
qed:16:	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑))

(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑))

Theorem

ordelordALTVD 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> MINIMIZE_WITH *" command.

1::	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   )
2:1:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Ord 𝐴   )
3:1:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ∈ 𝐴   )
4:2:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐴   )
5:2:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)   )
6:4,3:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ⊆ 𝐴   )
7:6,6,5:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)   )
8::	⊢ ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
9:8:	⊢ ∀𝑦((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
10:9:	⊢ ∀𝑦 ∈ 𝐴((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
11:10:	⊢ (∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
12:11:	⊢ ∀𝑥(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
13:12:	⊢ ∀𝑥 ∈ 𝐴(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
14:13:	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
15:14,5:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   )
16:4,15,3:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐵   )
17:16,7:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Ord 𝐵   )
qed:17:	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Theorem

equncomVD 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 = (𝐵 ∪ 𝐶)   ▶   𝐴 = (𝐵 ∪ 𝐶)   )
2::	⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
3:1,2:	⊢ (   𝐴 = (𝐵 ∪ 𝐶)   ▶   𝐴 = (𝐶 ∪ 𝐵)   )
4:3:	⊢ (𝐴 = (𝐵 ∪ 𝐶) → 𝐴 = (𝐶 ∪ 𝐵))
5::	⊢ (   𝐴 = (𝐶 ∪ 𝐵)   ▶   𝐴 = (𝐶 ∪ 𝐵)   )
6:5,2:	⊢ (   𝐴 = (𝐶 ∪ 𝐵)   ▶   𝐴 = (𝐵 ∪ 𝐶)   )
7:6:	⊢ (𝐴 = (𝐶 ∪ 𝐵) → 𝐴 = (𝐵 ∪ 𝐶))
8:4,7:	⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

(Contributed by Alan Sare, 17-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

Theorem

equncomiVD 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 = (𝐵 ∪ 𝐶)
qed:1:	⊢ 𝐴 = (𝐶 ∪ 𝐵)

(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝐴 = (𝐵 ∪ 𝐶)    ⇒   ⊢ 𝐴 = (𝐶 ∪ 𝐵)

Theorem

sucidALTVD 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 ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) is a semantic variation of the theorem (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))), which unifies with the set.mm reference definition (axiom) dford2 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

Theorem

sucidALT 40022

A set belongs to its successor. This proof was automatically derived from sucidALTVD 40021 using translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ suc 𝐴

Theorem

sucidVD 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

Theorem

imbi12VD 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::	⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   )
2::	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)    ▶   (𝜒 ↔ 𝜃)   )
3::	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜑 → 𝜒)   ▶   (𝜑 → 𝜒)   )
4:1,3:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜑 → 𝜒)   ▶   (𝜓 → 𝜒)   )
5:2,4:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜑 → 𝜒)   ▶   (𝜓 → 𝜃)   )
6:5:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)    ▶   ((𝜑 → 𝜒) → (𝜓 → 𝜃))   )
7::	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜓 → 𝜃)   ▶   (𝜓 → 𝜃)   )
8:1,7:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜓 → 𝜃)   ▶   (𝜑 → 𝜃)   )
9:2,8:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜓 → 𝜃)   ▶   (𝜑 → 𝜒)   )
10:9:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)    ▶   ((𝜓 → 𝜃) → (𝜑 → 𝜒))   )
11:6,10:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)    ▶   ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))   )
12:11:	⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))   )
qed:12:	⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))

Theorem

imbi13VD 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::	⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   )
2::	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)    ▶   (𝜒 ↔ 𝜃)   )
3::	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   (𝜏 ↔ 𝜂)   )
4:2,3:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂))   )
5:1,4:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)   ,   (𝜏 ↔ 𝜂)   ▶   ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))   )
6:5:	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ↔ 𝜃)    ▶   ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))   )
7:6:	⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))   )
qed:7:	⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))))

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))))))

Theorem

sbcim2gVD 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::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2::	⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   )
3:1,2:	⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))   )
4:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   )
5:3,4:	⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   )
6:5:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))   )
7::	⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   )
8:4,7:	⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))   )
9:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)))   )
10:8,9:	⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   )
11:10:	⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)))   )
12:6,11:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))   )
qed:12:	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

Theorem

sbcbiVD 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::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2::	⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)    ▶   ∀𝑥(𝜑 ↔ 𝜓)   )
3:1,2:	⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)    ▶   [𝐴 / 𝑥](𝜑 ↔ 𝜓)   )
4:1,3:	⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)    ▶   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)   )
5:4:	⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))   )
qed:5:	⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

Theorem

trsbcVD 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::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)   )
3:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴)   )
4:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝐴)   )
5:1,2,3,4:	⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))   )
6:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)))   )
7:5,6:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))   )
8::	⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
9:7,8:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
10::	⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
11:10:	⊢ ∀𝑥((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
12:1,11:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
13:9,12:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
14:13:	⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
15:14:	⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
16:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
17:15,16:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
18:17:	⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑧([𝐴 / 𝑥]∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
19:18:	⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑧[𝐴 / 𝑥]∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
20:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑧∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
21:19,20:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑧∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
22::	⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
23:21,22:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑧∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴)   )
24::	⊢ (Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
25:24:	⊢ ∀𝑥(Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
26:1,25:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]Tr 𝑥 ↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
27:23,26:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴)   )
qed:27:	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))

Theorem

truniALTVD 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::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴 Tr 𝑥   )
2::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   )
3:2:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ 𝑦   )
4:2:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑦 ∈ ∪ 𝐴   )
5:4:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   )
6::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   )
7:6:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑦 ∈ 𝑞   )
8:6:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑞 ∈ 𝐴   )
9:1,8:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   )
10:8,9:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   Tr 𝑞   )
11:3,7,10:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ 𝑞   )
12:11,8:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   )
13:12:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
14:13:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
15:14:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
16:5,15:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   )
17:16:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
18:17:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥    ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
19:18:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   Tr ∪ 𝐴   )
qed:19:	⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∪ 𝐴)

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Theorem

ee33VD 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 → (𝜒 → 𝜃)))
h2::	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
h3::	⊢ (𝜃 → (𝜏 → 𝜂))
4:1,3:	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
5:4:	⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
6:2,5:	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
7:6:	⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
8:7:	⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
qed:8:	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Theorem

trintALTVD 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::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴Tr 𝑥   )
2::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   )
3:2:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ 𝑦   )
4:2:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑦 ∈ ∩ 𝐴   )
5:4:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴𝑦 ∈ 𝑞   )
6:5:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞)   )
7::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑞 ∈ 𝐴   )
8:7,6:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑦 ∈ 𝑞   )
9:7,1:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   )
10:7,9:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   Tr 𝑞   )
11:10,3,8:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑧 ∈ 𝑞   )
12:11:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
13:12:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
14:13:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴𝑧 ∈ 𝑞   )
15:3,14:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ ∩ 𝐴   )
16:15:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
17:16:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
18:17:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   Tr ∩ 𝐴   )
qed:18:	⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∩ 𝐴)

(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Theorem

trintALT 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> MINIMIZE_WITH *" command. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Theorem

undif3VD 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 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)))
2::	⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
3:2:	⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
4:1,3:	⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
5::	⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
6:5:	⊢ (   𝑥 ∈ 𝐴   ▶   (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)   )
7:5:	⊢ (   𝑥 ∈ 𝐴   ▶   (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)   )
8:6,7:	⊢ (   𝑥 ∈ 𝐴   ▶   ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))   )
9:8:	⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
10::	⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   )
11:10:	⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   𝑥 ∈ 𝐵   )
12:10:	⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   ¬ 𝑥 ∈ 𝐶    )
13:11:	⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)   )
14:12:	⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)   )
15:13,14:	⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))   )
16:15:	⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
17:9,16:	⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
18::	⊢ (   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   )
19:18:	⊢ (   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   𝑥 ∈ 𝐴   )
20:18:	⊢ (   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   ¬ 𝑥 ∈ 𝐶    )
21:18:	⊢ (   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))   )
22:21:	⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
23::	⊢ (   (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)   ▶   (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)   )
24:23:	⊢ (   (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)   ▶   𝑥 ∈ 𝐴   )
25:24:	⊢ (   (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)   ▶   (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))   )
26:25:	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
27:10:	⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))   )
28:27:	⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
29::	⊢ (   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)   ▶   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)   )
30:29:	⊢ (   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)   ▶   𝑥 ∈ 𝐴   )
31:30:	⊢ (   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)   ▶   (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))   )
32:31:	⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
33:22,26:	⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
34:28,32:	⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
35:33,34:	⊢ ((((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∨ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
36::	⊢ ((((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∨ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
37:36,35:	⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
38:17,37:	⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
39::	⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
40:39:	⊢ (¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
41::	⊢ (¬ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴) ↔ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
42:40,41:	⊢ (¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
43::	⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ))
44:43,42:	⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴)))
45::	⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ ( 𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴)))
46:45,44:	⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ ( (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
47:4,38:	⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
48:46,47:	⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)))
49:48:	⊢ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)))
qed:49:	⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))

(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))

Theorem

sbcssgVD 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::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
3:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)   )
4:2,3:	⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷 ))   )
5:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷))   )
6:4,5:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
7:6:	⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
8:7:	⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷) )   )
9:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))   )
10:8,9:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷) )   )
11::	⊢ (𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
110:11:	⊢ ∀𝑥(𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
12:1,110:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))   )
13:10,12:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
14::	⊢ (⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ ∀ 𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))
15:13,14:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷)   )
qed:15:	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋ 𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷))

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷))

Theorem

csbingVD 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::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2::	⊢ (𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) }
20:2:	⊢ ∀𝑥(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}
30:1,20:	⊢ (   𝐴 ∈ 𝐵   ▶   [𝐴 / 𝑥](𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
3:1,30:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
4:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
5:3,4:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
6:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
7:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)   )
8:6,7:	⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷 ))   )
9:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷))   )
10:9,8:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
11:10:	⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
12:11:	⊢ (   𝐴 ∈ 𝐵   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   )
13:5,12:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   )
14::	⊢ (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) = { 𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}
15:13,14:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)   )
qed:15:	⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ( ⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Theorem

onfrALTlem5VD 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:	⊢ (𝑎 ∩ 𝑥) ∈ V
3:2:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ (𝑎 ∩ 𝑥) = ∅)
4:3:	⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ ¬ (𝑎 ∩ 𝑥) = ∅)
5::	⊢ ((𝑎 ∩ 𝑥) ≠ ∅ ↔ ¬ (𝑎 ∩ 𝑥 ) = ∅)
6:4,5:	⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ (𝑎 ∩ 𝑥) ≠ ∅)
7:2:	⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]¬ 𝑏 = ∅)
8::	⊢ (𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅)
9:8:	⊢ ∀𝑏(𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅)
10:2,9:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]¬ 𝑏 = ∅)
11:7,10:	⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅)
12:6,11:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅ ↔ ( 𝑎 ∩ 𝑥) ≠ ∅)
13:2:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥 ) ↔ (𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥))
14:12,13:	⊢ (([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅) ↔ ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅))
15:2:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅))
16:15,14:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) ↔ ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅))
17:2:	⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ( ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 ∩ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦)
18:2:	⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 = (𝑎 ∩ 𝑥)
19:2:	⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦 = 𝑦
20:18,19:	⊢ (⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 ∩ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦) = ((𝑎 ∩ 𝑥) ∩ 𝑦)
21:17,20:	⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = (( 𝑎 ∩ 𝑥) ∩ 𝑦)
22:2:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅ ↔ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ⦋(𝑎 ∩ 𝑥) / 𝑏⦌ ∅)
23:2:	⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌∅ = ∅
24:21,23:	⊢ (⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ⦋(𝑎 ∩ 𝑥) / 𝑏⦌∅ ↔ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
25:22,24:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅ ↔ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
26:2:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ↔ 𝑦 ∈ (𝑎 ∩ 𝑥))
27:25,26:	⊢ (([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ∧ [ (𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ (( 𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
28:2:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ∧ [(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅))
29:27,28:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
30:29:	⊢ ∀𝑦([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
31:30:	⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
32::	⊢ (∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ ))
33:31,32:	⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
34:2:	⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ ( 𝑏 ∩ 𝑦) = ∅))
35:33,34:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)
36::	⊢ (∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ ∃𝑦 (𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅))
37:36:	⊢ ∀𝑏(∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅))
38:2,37:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅))
39:35,38:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
40:16,39:	⊢ (([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
41:2:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅))
qed:40,41:	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥 )((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏 (𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))

Theorem

onfrALTlem4VD 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](𝑎 ∩ 𝑥) = ∅ ↔ ⦋ 𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅)
3:1:	⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌ 𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥)
4:1:	⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎
5:1:	⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦
6:4,5:	⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = ( 𝑎 ∩ 𝑦)
7:3,6:	⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦)
8:1:	⊢ ⦋𝑦 / 𝑥⦌∅ = ∅
9:7,8:	⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌ ∅ ↔ (𝑎 ∩ 𝑦) = ∅)
10:2,9:	⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎 ∩ 𝑦) = ∅)
11:1:	⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎)
12:11,10:	⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥]( 𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
13:1:	⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅))
qed:13,12:	⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))

Theorem

onfrALTlem3VD 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 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   )
3:2:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   𝑥 ∈ 𝑎   )
4:1:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ⊆ On   )
5:3,4:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   𝑥 ∈ On   )
6:5:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   Ord 𝑥   )
7:6:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶    E We 𝑥   )
8::	⊢ (𝑎 ∩ 𝑥) ⊆ 𝑥
9:7,8:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶    E We (𝑎 ∩ 𝑥)   )
10:9:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶    E Fr (𝑎 ∩ 𝑥)   )
11:10:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∀𝑏((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅)   )
12::	⊢ 𝑥 ∈ V
13:12,8:	⊢ (𝑎 ∩ 𝑥) ∈ V
14:13,11:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   [(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅)   )
15::	⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)( (𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
16:14,15:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ ( 𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)   )
17::	⊢ (𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥)
18:2:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ¬ (𝑎 ∩ 𝑥) = ∅   )
19:18:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑎 ∩ 𝑥) ≠ ∅   )
20:17,19:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅)   )
qed:16,20:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅   )

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅   )

Theorem

simplbi2comtVD 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 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜑 ↔ ( 𝜓 ∧ 𝜒))   )
2:1:	⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   ((𝜓 ∧ 𝜒 ) → 𝜑)   )
3:2:	⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜓 → (𝜒 → 𝜑))   )
4:3:	⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜒 → (𝜓 → 𝜑))   )
qed:4:	⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑)))

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑)))

Theorem

onfrALTlem2VD 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 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   )
2:1:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ (𝑎 ∩ 𝑦)   )
3:2:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ 𝑎   )
4::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
5::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   )
6:5:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   𝑥 ∈ 𝑎   )
7:4:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ⊆ On   )
8:6,7:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   𝑥 ∈ On   )
9:8:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   Ord 𝑥   )
10:9:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   Tr 𝑥   )
11:1:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑦 ∈ (𝑎 ∩ 𝑥)   )
12:11:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑦 ∈ 𝑥   )
13:2:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ 𝑦   )
14:10,12,13:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ 𝑥   )
15:3,14:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ (𝑎 ∩ 𝑥)   )
16:13,15:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)   )
17:16:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦))   )
18:17:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   ∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦))   )
19:18:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦)   )
20::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)   )
21:20:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅   )
22:19,21:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑎 ∩ 𝑦) = ∅   )
23:20:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   𝑦 ∈ (𝑎 ∩ 𝑥)   )
24:23:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   𝑦 ∈ 𝑎   )
25:22,24:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)   )
26:25:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))   )
27:26:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∀𝑦((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥 ) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))   )
28:27:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥 ) ∩ 𝑦) = ∅) → ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))   )
29::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅   )
30:29:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)   )
31:28,30:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)   )
qed:31:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )

Theorem

onfrALTlem1VD 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 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
2:1:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
3:2:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)    )
4::	⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅ ) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
5:4:	⊢ ∀𝑦([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
6:5:	⊢ (∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
7:3,6:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)   )
8::	⊢ (∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦( 𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
qed:7,8:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )

Theorem

onfrALTVD 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 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
2::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
3:1:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶    (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
4:2:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶    ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
5::	⊢ ((𝑎 ∩ 𝑥) = ∅ ∨ ¬ (𝑎 ∩ 𝑥) = ∅)
6:5,4,3:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶    ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
7:6:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
8:7:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
9:8:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
10::	⊢ (𝑎 ≠ ∅ ↔ ∃𝑥𝑥 ∈ 𝑎)
11:9,10:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
12::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
13:12:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ≠ ∅   )
14:13,11:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
15:14:	⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)
16:15:	⊢ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)
qed:16:	⊢ E Fr On

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ E Fr On

Theorem

csbeq2gVD 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::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥𝐵 = 𝐶 → [𝐴 / 𝑥] 𝐵 = 𝐶)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   )
4:2,3:	⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥 ⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   )
qed:4:	⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌ 𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

Theorem

csbsngVD 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::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   )
4:3:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
5:2,4:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
6:5:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
7:6:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
8:1:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
9:7,8:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
10::	⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
11:10:	⊢ ∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
12:1,11:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = ⦋ 𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
13:9,12:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { 𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
14::	⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}
15:13,14:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { ⦋𝐴 / 𝑥⦌𝐵}   )
qed:15:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋ 𝐴 / 𝑥⦌𝐵})

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})

Theorem

csbxpgVD 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::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑤 = 𝑤   )
4:3:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
5:2,4:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
6:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
7:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   )
8:7:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
9:6,8:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
10:5,9:	⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))   )
11:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶))   )
12:10,11:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))   )
13:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑧 = ⟨𝑤   ,    𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩)   )
14:12,13:	⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
15:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   )
16:14,15:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
17:16:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
18:17:	⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
19:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   )
20:18,19:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
21:20:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑤([𝐴 / 𝑥]∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
22:21:	⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
23:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦 (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   )
24:22,23:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
25:24:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑧([𝐴 / 𝑥]∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
26:25:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}    )
27:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥] ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}   )
28:26,27:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}    )
29::	⊢ {⟨𝑤   ,   𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
30::	⊢ (𝐵 × 𝐶) = {⟨𝑤   ,   𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
31:29,30:	⊢ (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤 , 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
32:31:	⊢ ∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
33:1,32:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}   )
34:28,33:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}   )
35::	⊢ {⟨𝑤   ,   𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
36::	⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = { ⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)}
37:35,36:	⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
38:34,37:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶)   )
qed:38:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))

Theorem

csbresgVD 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::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌V = V   )
3:2:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   )
4:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)   )
5:3,4:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   )
6:5:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   )
7:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))   )
8:6,7:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   )
9::	⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
10:9:	⊢ ∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
11:1,10:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V))   )
12:8,11:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   )
13::	⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))
14:12,13:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)   )
qed:14:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶))

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶))

Theorem

csbrngVD 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::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤   ,   𝑦⟩ ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌⟨𝑤   ,   𝑦⟩ = ⟨𝑤, 𝑦⟩   )
4:3:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌⟨𝑤   ,   𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
5:2,4:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤   ,   𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
6:5:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑤([𝐴 / 𝑥]⟨𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
7:6:	⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
8:1:	⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵)   )
9:7,8:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤⟨𝑤    ,   𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
10:9:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]∃𝑤 ⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
11:10:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨ 𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
12:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤 ⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   )
13:11,12:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤 ⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
14::	⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤   ,   𝑦⟩ ∈ 𝐵}
15:14:	⊢ ∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤   ,   𝑦⟩ ∈ 𝐵}
16:1,15:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   )
17:13,16:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
18::	⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤    ,   𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
19:17,18:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋ 𝐴 / 𝑥⦌𝐵   )
qed:19:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)

Theorem

csbima12gALTVD 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::	⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   )
2:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ( ⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
3:2:	⊢ (   𝐴 ∈ 𝐶   ▶    ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
4:1:	⊢ (   𝐴 ∈ 𝐶   ▶    ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵)   )
5:3,4:	⊢ (   𝐴 ∈ 𝐶   ▶    ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
6::	⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
7:6:	⊢ ∀𝑥(𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
8:1,7:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋ 𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)   )
9:5,8:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
10::	⊢ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)
11:9,10:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ( ⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)   )
qed:11:	⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋ 𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))

Theorem

csbunigVD 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::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
4:2,3:	⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   )
5:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵))   )
6:4,5:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   )
7:6:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   )
8:7:	⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   )
9:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))   )
10:8,9:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   )
11:10:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑧([𝐴 / 𝑥]∃𝑦( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   )
12:11:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑧 ∣ [𝐴 / 𝑥]∃𝑦( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}   )
13:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}    )
14:12,13:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}   )
15::	⊢ ∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
16:15:	⊢ ∀𝑥∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
17:1,16:	⊢ (   𝐴 ∈ 𝑉   ▶   [𝐴 / 𝑥]∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}   )
18:1,17:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌∪ 𝐵 = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}   )
19:14,18:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}   )
20::	⊢ ∪ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}
21:19,20:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵   )
qed:21:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)

Theorem

csbfv12gALTVD 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::	⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   )
2:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦} = { 𝑦}   )
3:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵 }) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵})   )
4:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { ⦋𝐴 / 𝑥⦌𝐵}   )
5:4:	⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   )
6:3,5:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵 }) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   )
7:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦})   )
8:6,2:	⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌(𝐹 “ { 𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
9:7,8:	⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})    )
10:9:	⊢ (   𝐴 ∈ 𝐶   ▶   ∀𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
11:10:	⊢ (   𝐴 ∈ 𝐶   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
12:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}   )
13:11,12:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦 }}   )
14:13:	⊢ (   𝐴 ∈ 𝐶   ▶   ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
15:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
16:14,15:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
17::	⊢ (𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
18:17:	⊢ ∀𝑥(𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵 }) = {𝑦}}
19:1,18:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
20:16,19:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
21::	⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}
22:20,21:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)   )
qed:22:	⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))

Theorem

con5VD 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)   ▶   (¬ 𝜓 → 𝜑)   )
3:2:	⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → ¬ ¬ 𝜓 )   )
4::	⊢ (𝜓 ↔ ¬ ¬ 𝜓)
5:3,4:	⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → 𝜓)   )
qed:5:	⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))

(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))

Theorem

relopabVD 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 = 𝑦 ↔ 𝑦 = 𝑣)
91:90:	⊢ (∃𝑣𝑣 = 𝑦 ↔ ∃𝑣𝑦 = 𝑣)
92::	⊢ ∃𝑣𝑣 = 𝑦
10:91,92:	⊢ ∃𝑣𝑦 = 𝑣
11:9,10:	⊢ ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
12:11:	⊢ (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
13::	⊢ (∃𝑣(𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢 , 𝑣⟩) → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩))
14:12,13:	⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
15:14:	⊢ (∃𝑢𝑥 = 𝑢 → ∃𝑢(𝑧 = ⟨𝑥   ,   𝑦 ⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩))
150::	⊢ (𝑢 = 𝑥 ↔ 𝑥 = 𝑢)
151:150:	⊢ (∃𝑢𝑢 = 𝑥 ↔ ∃𝑢𝑥 = 𝑢)
152::	⊢ ∃𝑢𝑢 = 𝑥
16:151,152:	⊢ ∃𝑢𝑥 = 𝑢
17:15,16:	⊢ ∃𝑢(𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑣𝑧 = ⟨ 𝑢, 𝑣⟩)
18:17:	⊢ (𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢∃𝑣𝑧 = ⟨ 𝑢, 𝑣⟩)
19:18:	⊢ (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑦∃𝑢 ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
20::	⊢ (∃𝑦∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
21:19,20:	⊢ (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
22:21:	⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑥 ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
23::	⊢ (∃𝑥∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
24:22,23:	⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢 ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
25:24:	⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩}
26::	⊢ 𝑥 ∈ V
27::	⊢ 𝑦 ∈ V
28:26,27:	⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
29:28:	⊢ (𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ (𝑧 = ⟨𝑥   ,   𝑦 ⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
30:29:	⊢ (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
31:30:	⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑥 ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
32:31:	⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} = { 𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
320:25,32:	⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩}
33::	⊢ 𝑢 ∈ V
34::	⊢ 𝑣 ∈ V
35:33,34:	⊢ (𝑢 ∈ V ∧ 𝑣 ∈ V)
36:35:	⊢ (𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ (𝑧 = ⟨𝑢   ,   𝑣 ⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
37:36:	⊢ (∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
38:37:	⊢ (∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑢 ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
39:38:	⊢ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩} = { 𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
40:320,39:	⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
41::	⊢ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) }
42::	⊢ {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) }
43:40,41,42:	⊢ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
44::	⊢ {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = (V × V)
45:43,44:	⊢ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} ⊆ (V × V)
46:28:	⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
47:46:	⊢ {⟨𝑥   ,   𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
48:45,47:	⊢ {⟨𝑥   ,   𝑦⟩ ∣ 𝜑} ⊆ (V × V)
qed:48:	⊢ Rel {⟨𝑥   ,   𝑦⟩ ∣ 𝜑}

(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Theorem

19.41rgVD 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 → (𝜑 ∧ 𝜓)))
2:1:	⊢ ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → ( 𝜑 ∧ 𝜓))))
3:2:	⊢ ∀𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))))
4:3:	⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑 ∧ 𝜓))))
5::	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   ∀𝑥(𝜓 → ∀𝑥𝜓)   )
6:4,5:	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑 ∧ 𝜓)))   )
7::	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ,   ∀𝑥𝜓   ▶    ∀𝑥𝜓   )
8:6,7:	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ,   ∀𝑥𝜓   ▶    ∀𝑥(𝜑 → (𝜑 ∧ 𝜓))   )
9:8:	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ,   ∀𝑥𝜓   ▶    (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))   )
10:9:	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))   )
11:5:	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀ 𝑥𝜓)   )
12:10,11:	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ( ∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))   )
13:12:	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))   )
14:13:	⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥 𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))   )
qed:14:	⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))

⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))

Theorem

2pm13.193VD 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 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2:1:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
3:2:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   )
4:1:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
5:3,4:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
6:5:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
7:6:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
8:2:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   )
9:7,8:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)   )
10:9:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝜑 ∧ 𝑦 = 𝑣)   )
11:10:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝜑   )
12:2,11:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   )
13:12:	⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
14::	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   )
15:14:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
16:15:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
17:14:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑    )
18:16,17:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ( 𝜑 ∧ 𝑦 = 𝑣)   )
19:18:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)   )
20:15:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
21:19:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
22:20,21:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
23:22:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
24:23:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
25:15,24:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
26:25:	⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
qed:13,26:	⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Theorem

hbimpgVD 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))   ▶   ∀𝑥(¬ 𝜑 → ∀𝑥¬ 𝜑)   )
5:4:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   (¬ 𝜑 → ∀𝑥¬ 𝜑)   )
6:3,5:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)), ¬ 𝜑   ▶   ∀𝑥¬ 𝜑   )
7::	⊢ (¬ 𝜑 → (𝜑 → 𝜓))
8:7:	⊢ (∀𝑥¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
9:6,8:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)), ¬ 𝜑   ▶   ∀𝑥(𝜑 → 𝜓)   )
10:9:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   (¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))   )
11::	⊢ (𝜓 → (𝜑 → 𝜓))
12:11:	⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓))
13:1:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ∀𝑥(𝜓 → ∀𝑥𝜓)   )
14:13:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   (𝜓 → ∀𝑥𝜓)   )
15:14,12:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   (𝜓 → ∀𝑥(𝜑 → 𝜓))   )
16:10,15:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ((¬ 𝜑 ∨ 𝜓) → ∀𝑥(𝜑 → 𝜓))   )
17::	⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
18:16,17:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))   )
19::	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥( 𝜑 → ∀𝑥𝜑))
20::	⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ∀𝑥∀𝑥( 𝜓 → ∀𝑥𝜓))
21:19,20:	⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥(∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)))
22:21,18:	⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))   )
qed:22:	⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))

⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))

Theorem

hbalgVD 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 → ∀𝑦∀𝑥𝜑)   )
3::	⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
4:2,3:	⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)   )
5::	⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦( 𝜑 → ∀𝑥𝜑))
6:5,4:	⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∀ 𝑦𝜑 → ∀𝑥∀𝑦𝜑)   )
qed:6:	⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦 𝜑 → ∀𝑥∀𝑦𝜑))

⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Theorem

hbexgVD 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::	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(𝜑 → ∀𝑥𝜑)   )
2:1:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 ∀𝑥(𝜑 → ∀𝑥𝜑)   )
3:2:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (𝜑 → ∀𝑥𝜑)   )
4:3:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (¬ 𝜑 → ∀𝑥¬ 𝜑)   )
5::	⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦 ∀𝑥(𝜑 → ∀𝑥𝜑))
6::	⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦 ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
7:5:	⊢ (∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
8:5,6,7:	⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦 ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
9:8,4:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 ∀𝑥(¬ 𝜑 → ∀𝑥¬ 𝜑)   )
10:9:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(¬ 𝜑 → ∀𝑥¬ 𝜑)   )
11:10:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 (¬ 𝜑 → ∀𝑥¬ 𝜑)   )
12:11:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 (∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑)   )
13:12:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀ 𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑)   )
14::	⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥 ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
15:13,14:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑)   )
16:15:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
17:16:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
18::	⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦¬ 𝜑)
19:17,18:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
20:18:	⊢ (∀𝑥∃𝑦𝜑 ↔ ∀𝑥¬ ∀𝑦¬ 𝜑)
21:19,20:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
22:8,21:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
23:14,22:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
qed:23:	⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )

⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Theorem

ax6e2eqVD 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 = 𝑢 ∧ 𝑦 = 𝑢)   )
6:5:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))   )
7:6:	⊢ (∀𝑥𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
8:7:	⊢ (∀𝑥∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
9::	⊢ (∀𝑥𝑥 = 𝑦 ↔ ∀𝑥∀𝑥𝑥 = 𝑦)
10:8,9:	⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
11:1,10:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))   )
12:11:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (∃𝑥𝑥 = 𝑢 → ∃𝑥 (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))   )
13::	⊢ ∃𝑥𝑥 = 𝑢
14:13,12:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢 )   )
140:14:	⊢ (∀𝑥𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) )
141:140:	⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢))
15:1,141:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   )
16:1,15:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   )
17:16:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   )
18:17:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   )
19::	⊢ (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   )
20::	⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   )
21:20:	⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   𝑦 = 𝑢    )
22:19,21:	⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   𝑦 = 𝑣    )
23:20:	⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   𝑥 = 𝑢    )
24:22,23:	⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
25:24:	⊢ (   𝑢 = 𝑣   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   )
26:25:	⊢ (   𝑢 = 𝑣   ▶   ∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   )
27:26:	⊢ (   𝑢 = 𝑣   ▶   (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   )
28:27:	⊢ (   𝑢 = 𝑣   ▶   ∀𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   )
29:28:	⊢ (   𝑢 = 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   )
30:29:	⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢 ) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
31:18,30:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (𝑢 = 𝑣 → ∃𝑥∃𝑦 (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   )
qed:31:	⊢ (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))

⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))

Theorem

ax6e2ndVD 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 = 𝑢 ∧ 𝑦 = 𝑣))
7:6:	⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦 (∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
8:4,7:	⊢ ∃𝑦(∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣)
9::	⊢ (𝑧 = 𝑣 → ∀𝑥𝑧 = 𝑣)
10::	⊢ (𝑦 = 𝑣 → ∀𝑧𝑦 = 𝑣)
11::	⊢ (   𝑧 = 𝑦   ▶   𝑧 = 𝑦   )
12:11:	⊢ (   𝑧 = 𝑦   ▶   (𝑧 = 𝑣 ↔ 𝑦 = 𝑣)   )
120:11:	⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
13:9,10,120:	⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥𝑦 = 𝑣))
14::	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ¬ ∀𝑥𝑥 = 𝑦   )
15:14,13:	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣)   )
16:15:	⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥𝑦 = 𝑣))
17:16:	⊢ (∀𝑥¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥𝑦 = 𝑣))
18::	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦 )
19:17,18:	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀ 𝑥𝑦 = 𝑣))
20:14,19:	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥(𝑦 = 𝑣 → ∀𝑥𝑦 = 𝑣)   )
21:20:	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ((∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   )
22:21:	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ((∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
23:22:	⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
24::	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦 )
25:23,24:	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
26:14,25:	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ∀𝑦((∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   )
27:26:	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   (∃𝑦(∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   )
28:8,27:	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
29:28:	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
qed:29:	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Theorem

ax6e2ndeqVD 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::	⊢ (   𝑢 ≠ 𝑣   ▶   𝑢 ≠ 𝑣   )
2::	⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
3:2:	⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   𝑥 = 𝑢   )
4:1,3:	⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   𝑥 ≠ 𝑣   )
5:2:	⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   𝑦 = 𝑣   )
6:4,5:	⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   𝑥 ≠ 𝑦   )
7::	⊢ (∀𝑥𝑥 = 𝑦 → 𝑥 = 𝑦)
8:7:	⊢ (¬ 𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
9::	⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
10:8,9:	⊢ (𝑥 ≠ 𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
11:6,10:	⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶    ¬ ∀𝑥𝑥 = 𝑦   )
12:11:	⊢ (   𝑢 ≠ 𝑣   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
13:12:	⊢ (   𝑢 ≠ 𝑣   ▶   ∀𝑥((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
14:13:	⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥¬ ∀𝑥𝑥 = 𝑦)   )
15::	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦 )
19:15:	⊢ (∃𝑥¬ ∀𝑥𝑥 = 𝑦 ↔ ¬ ∀𝑥𝑥 = 𝑦)
20:14,19:	⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
21:20:	⊢ (   𝑢 ≠ 𝑣   ▶   ∀𝑦(∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
22:21:	⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦)   )
23::	⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ ∃ 𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
24:22,23:	⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦)   )
25::	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦 )
26:25:	⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∃𝑦∀𝑦¬ ∀𝑥𝑥 = 𝑦)
260::	⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦∀𝑦¬ ∀𝑥𝑥 = 𝑦)
27:260:	⊢ (∃𝑦∀𝑦¬ ∀𝑥𝑥 = 𝑦 ↔ ∀𝑦¬ ∀𝑥𝑥 = 𝑦)
270:26,27:	⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥 𝑥 = 𝑦)
28::	⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦 )
29:270,28:	⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦 )
30:24,29:	⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   )
31:30:	⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣))   )
32:31:	⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
33::	⊢ (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   )
34:33:	⊢ (   𝑢 = 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑢 = 𝑣)   )
35:34:	⊢ (   𝑢 = 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣))   )
36:35:	⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
37::	⊢ (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣)
38:32,36,37:	⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ( ¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
39::	⊢ (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦 (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
40::	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
41:40:	⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃ 𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
42::	⊢ (∀𝑥𝑥 = 𝑦 ∨ ¬ ∀𝑥𝑥 = 𝑦)
43:39,41,42:	⊢ (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ))
44:40,43:	⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ∃𝑥 ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
qed:38,44:	⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥 ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Theorem

2sb5ndVD 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 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
2:1:	⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
3::	⊢ ([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
4:3:	⊢ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
5:4:	⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥] ∀𝑦[𝑣 / 𝑦]𝜑)
6::	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ¬ ∀𝑥𝑥 = 𝑦   )
7::	⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑥𝑥 = 𝑦)
8:7:	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑦𝑦 = 𝑥)
9:6,8:	⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ¬ ∀𝑦𝑦 = 𝑥   )
10:9:	⊢ ([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀ 𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
11:5,10:	⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
12:11:	⊢ (¬ ∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
13::	⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
14::	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥𝑥 = 𝑦   )
15:14:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (∀𝑥[𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
16:13,15:	⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦 ]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
17:16:	⊢ (∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦] 𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
19:12,17:	⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
20:19:	⊢ (∃𝑦((