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Theorem vtoclef 3024
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 2962 . 2  |-  E. x  x  =  A
3 vtoclef.1 . . 3  |-  F/ x ph
4 vtoclef.3 . . 3  |-  ( x  =  A  ->  ph )
53, 4exlimi 1821 . 2  |-  ( E. x  x  =  A  ->  ph )
62, 5ax-mp 8 1  |-  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550   F/wnf 1553    = wceq 1652    e. wcel 1725   _Vcvv 2956
This theorem is referenced by:  nn0ind-raph  10370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-v 2958
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