Theorem vtocl2gf 1385

Description: Implicit substitution of a class for a set variable.

Hypotheses

Ref	Expression
vtocl2gf.1	⊢ (ψ → ∀xψ)
vtocl2gf.2	⊢ (χ → ∀yχ)
vtocl2gf.3	⊢ (x = A → (φ ↔ ψ))
vtocl2gf.4	⊢ (y = B → (ψ ↔ χ))
vtocl2gf.5	⊢ φ

Assertion

Ref	Expression
vtocl2gf	⊢ ((A ∈ C ∧ B ∈ D) → χ)

Distinct variable group(s):   x,A   y,A   y,B

Proof of Theorem vtocl2gf

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (z ∈ B → ∀y z ∈ B)
2		ax-17 925	. . . . . 6 ⊢ (A ∈ V → ∀y A ∈ V)
3		vtocl2gf.2	. . . . . 6 ⊢ (χ → ∀yχ)
4	2, 3	hbim 702	. . . . 5 ⊢ ((A ∈ V → χ) → ∀y(A ∈ V → χ))
5		vtocl2gf.4	. . . . . 6 ⊢ (y = B → (ψ ↔ χ))
6	5	imbi2d 464	. . . . 5 ⊢ (y = B → ((A ∈ V → ψ) ↔ (A ∈ V → χ)))
7		ax-17 925	. . . . . 6 ⊢ (z ∈ A → ∀x z ∈ A)
8		vtocl2gf.1	. . . . . 6 ⊢ (ψ → ∀xψ)
9		vtocl2gf.3	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
10		vtocl2gf.5	. . . . . 6 ⊢ φ
11	7, 8, 9, 10	vtoclgf 1382	. . . . 5 ⊢ (A ∈ V → ψ)
12	1, 4, 6, 11	vtoclgf 1382	. . . 4 ⊢ (B ∈ D → (A ∈ V → χ))
13	12	com12 13	. . 3 ⊢ (A ∈ V → (B ∈ D → χ))
14	13	imp 277	. 2 ⊢ ((A ∈ V ∧ B ∈ D) → χ)
15		elisset 1354	. 2 ⊢ (A ∈ C → A ∈ V)
16	14, 15	sylan 343	1 ⊢ ((A ∈ C ∧ B ∈ D) → χ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  vtocl2g 1386

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

metamath.org