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Theorem vtoclef 2857
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 2795 . 2  |-  E. x  x  =  A
3 vtoclef.1 . . 3  |-  F/ x ph
4 vtoclef.3 . . 3  |-  ( x  =  A  ->  ph )
53, 4exlimi 1802 . 2  |-  ( E. x  x  =  A  ->  ph )
62, 5ax-mp 10 1  |-  ph
Colors of variables: wff set class
Syntax hints:    -> wi 6   E.wex 1529   F/wnf 1532    = wceq 1624    e. wcel 1685   _Vcvv 2789
This theorem is referenced by:  nn0ind-raph  10107
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-11 1716  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-v 2791
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