Theorem vtoclef 2853

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

Hypotheses

Ref	Expression
vtoclef.1	|- F/ x ph
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 2785	. 2 |- E. x x = A
3		vtoclef.1	. . 3 |- F/ x ph
4		vtoclef.3	. . 3 |- ( x = A -> ph )
5	3, 4	exlimi 1618	. 2 |- ( E. x x = A -> ph )
6	2, 5	ax-mp 5	1 |- ph

Syntax hints:   -> wi 4   = wceq 1373   F/wnf 1484   E.wex 1516   e. wcel 2178   _Vcvv 2776

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778

This theorem is referenced by:  nn0ind-raph  9525