Theorem vtoclef 2927

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)

Hypotheses

Ref	Expression
vtoclef.1	⊢ Ⅎxφ
vtoclef.2	⊢ A ∈ V
vtoclef.3	⊢ (x = A → φ)

Assertion

Ref	Expression
vtoclef	⊢ φ

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem vtoclef

Step	Hyp	Ref	Expression
1		vtoclef.2	. . 3 ⊢ A ∈ V
2	1	isseti 2865	. 2 ⊢ ∃x x = A
3		vtoclef.1	. . 3 ⊢ Ⅎxφ
4		vtoclef.3	. . 3 ⊢ (x = A → φ)
5	3, 4	exlimi 1803	. 2 ⊢ (∃x x = A → φ)
6	2, 5	ax-mp 5	1 ⊢ φ

Syntax hints:   → wi 4  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Vcvv 2859

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 2861

This theorem is referenced by: (None)