Theorem vtocleg 2797

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)

Hypothesis

Ref	Expression
vtocleg.1	|- ( x = A -> ph )

Assertion

Ref	Expression
vtocleg	|- ( A e. V -> ph )

Distinct variable groups:   x, A   ph, x

Allowed substitution hint:   V( x)

Proof of Theorem vtocleg

Step	Hyp	Ref	Expression
1		elisset 2740	. 2 |- ( A e. V -> E. x x = A )
2		vtocleg.1	. . 3 |- ( x = A -> ph )
3	2	exlimiv 1586	. 2 |- ( E. x x = A -> ph )
4	1, 3	syl 14	1 |- ( A e. V -> ph )

Syntax hints:   -> wi 4   = wceq 1343   E.wex 1480   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  vtocle  2800  spsbc  2962  prexg  4189  funimaexglem  5271  eloprabga  5929  cc3  7209  bj-prexg  13793