Theorem vtoclef 2803

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 2738	. 2 |- E. x x = A
3		vtoclef.1	. . 3 |- F/ x ph
4		vtoclef.3	. . 3 |- ( x = A -> ph )
5	3, 4	exlimi 1587	. 2 |- ( E. x x = A -> ph )
6	2, 5	ax-mp 5	1 |- ph

Syntax hints:   -> wi 4   = wceq 1348   F/wnf 1453   E.wex 1485   e. wcel 2141   _Vcvv 2730

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  nn0ind-raph  9329