Theorem vtocl2ga 2923

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
vtocl2ga.1	⊢ (x = A → (φ ↔ ψ))
vtocl2ga.2	⊢ (y = B → (ψ ↔ χ))
vtocl2ga.3	⊢ ((x ∈ C ∧ y ∈ D) → φ)

Assertion

Ref	Expression
vtocl2ga	⊢ ((A ∈ C ∧ B ∈ D) → χ)

Distinct variable groups:   x,y,A   y,B   x,C,y   x,D,y   ψ,x   χ,y

Allowed substitution hints:   φ(x,y)   ψ(y)   χ(x)   B(x)

Proof of Theorem vtocl2ga

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxA
2		nfcv 2490	. 2 ⊢ ℲyA
3		nfcv 2490	. 2 ⊢ ℲyB
4		nfv 1619	. 2 ⊢ Ⅎxψ
5		nfv 1619	. 2 ⊢ Ⅎyχ
6		vtocl2ga.1	. 2 ⊢ (x = A → (φ ↔ ψ))
7		vtocl2ga.2	. 2 ⊢ (y = B → (ψ ↔ χ))
8		vtocl2ga.3	. 2 ⊢ ((x ∈ C ∧ y ∈ D) → φ)
9	1, 2, 3, 4, 5, 6, 7, 8	vtocl2gaf 2922	1 ⊢ ((A ∈ C ∧ B ∈ D) → χ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  f1fveq  5474  caovcan  5629