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Theorem vtoclef 2984
Description: Implicit substitution of a class for a set 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
StepHypRef Expression
1 vtoclef.2 . . 3  |-  A  e. 
_V
21isseti 2922 . 2  |-  E. x  x  =  A
3 vtoclef.1 . . 3  |-  F/ x ph
4 vtoclef.3 . . 3  |-  ( x  =  A  ->  ph )
53, 4exlimi 1817 . 2  |-  ( E. x  x  =  A  ->  ph )
62, 5ax-mp 8 1  |-  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1547   F/wnf 1550    = wceq 1649    e. wcel 1721   _Vcvv 2916
This theorem is referenced by:  nn0ind-raph  10326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-11 1757  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-v 2918
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