Theorem vtocleg 2878

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 2818	. 2 |- ( A e. V -> E. x x = A )
2		vtocleg.1	. . 3 |- ( x = A -> ph )
3	2	exlimiv 1647	. 2 |- ( E. x x = A -> ph )
4	1, 3	syl 14	1 |- ( A e. V -> ph )

Syntax hints:   -> wi 4   = wceq 1398   E.wex 1541   e. wcel 2202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2805

This theorem is referenced by:  vtocle  2881  spsbc  3044  prexg  4307  funimaexglem  5420  eloprabga  6118  cc3  7530  bj-prexg  16610