Theorem vtoclef 3191

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 3124	. 2 |- E. x x = A
3		vtoclef.1	. . 3 |- F/ x ph
4		vtoclef.3	. . 3 |- ( x = A -> ph )
5	3, 4	exlimi 1859	. 2 |- ( E. x x = A -> ph )
6	2, 5	ax-mp 5	1 |- ph

Syntax hints:   -> wi 4   = wceq 1379   E.wex 1596   F/wnf 1599   e. wcel 1767   _Vcvv 3118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3120

This theorem is referenced by:  nn0ind-raph  10973