Theorem vtocl2gf 1852

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 groups:   x,A   y,A   y,B

Proof of Theorem vtocl2gf

Step	Hyp	Ref	Expression
1		ax-17 973	. . . 4 ⊢ (z ∈ B → ∀y z ∈ B)
2		ax-17 973	. . . . 5 ⊢ (A ∈ V → ∀y A ∈ V)
3		vtocl2gf.2	. . . . 5 ⊢ (χ → ∀yχ)
4	2, 3	hbim 1009	. . . 4 ⊢ ((A ∈ V → χ) → ∀y(A ∈ V → χ))
5		vtocl2gf.4	. . . . 5 ⊢ (y = B → (ψ ↔ χ))
6	5	imbi2d 614	. . . 4 ⊢ (y = B → ((A ∈ V → ψ) ↔ (A ∈ V → χ)))
7		ax-17 973	. . . . 5 ⊢ (z ∈ A → ∀x z ∈ A)
8		vtocl2gf.1	. . . . 5 ⊢ (ψ → ∀xψ)
9		vtocl2gf.3	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
10		vtocl2gf.5	. . . . 5 ⊢ φ
11	7, 8, 9, 10	vtoclgf 1849	. . . 4 ⊢ (A ∈ V → ψ)
12	1, 4, 6, 11	vtoclgf 1849	. . 3 ⊢ (B ∈ D → (A ∈ V → χ))
13	12	impcom 351	. 2 ⊢ ((A ∈ V ⋀ B ∈ D) → χ)
14		elisset 1820	. 2 ⊢ (A ∈ C → A ∈ V)
15	13, 14	sylan 450	1 ⊢ ((A ∈ C ⋀ B ∈ D) → χ)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  Vcvv 1814

This theorem is referenced by:  vtocl2g 1853

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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