Theorem vtocl 2910

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)

Hypotheses

Ref	Expression
vtocl.1	⊢ A ∈ V
vtocl.2	⊢ (x = A → (φ ↔ ψ))
vtocl.3	⊢ φ

Assertion

Ref	Expression
vtocl	⊢ ψ

Distinct variable groups:   x,A   ψ,x

Allowed substitution hint:   φ(x)

Proof of Theorem vtocl

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxψ
2		vtocl.1	. 2 ⊢ A ∈ V
3		vtocl.2	. 2 ⊢ (x = A → (φ ↔ ψ))
4		vtocl.3	. 2 ⊢ φ
5	1, 2, 3, 4	vtoclf 2909	1 ⊢ ψ

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860

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-11 1746  ax-ext 2334

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

This theorem is referenced by:  vtoclb  2913  caovcan  5629  enprmapc  6084