Theorem vtoclef 2927

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

Hypotheses

Ref	Expression
vtoclef.1	|- F/xph
vtoclef.2	|- A e. _V
vtoclef.3	|- (x = A -> ph)

Assertion

Ref	Expression
vtoclef	|- ph

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem vtoclef

Step	Hyp	Ref	Expression
1		vtoclef.2	. . 3 |- A e. _V
2	1	isseti 2865	. 2 |- E.x x = A
3		vtoclef.1	. . 3 |- F/xph
4		vtoclef.3	. . 3 |- (x = A -> ph)
5	3, 4	exlimi 1803	. 2 |- (E.x x = A -> ph)
6	2, 5	ax-mp 5	1 |- ph

Syntax hints:   -> wi 4  E.wex 1541   F/wnf 1544   = wceq 1642   e. 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)