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Theorem vtoclef 3122
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
StepHypRef Expression
1 vtoclef.2 . . 3  |-  A  e. 
_V
21isseti 3051 . 2  |-  E. x  x  =  A
3 vtoclef.1 . . 3  |-  F/ x ph
4 vtoclef.3 . . 3  |-  ( x  =  A  ->  ph )
53, 4exlimi 1995 . 2  |-  ( E. x  x  =  A  ->  ph )
62, 5ax-mp 5 1  |-  ph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444   E.wex 1663   F/wnf 1667    e. wcel 1887   _Vcvv 3045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3047
This theorem is referenced by:  nn0ind-raph  11035  finxpreclem2  31782
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