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Theorem vtoclf 2802
Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1767. (Contributed by NM, 30-Aug-1993.)
Hypotheses
Ref Expression
vtoclf.1  |-  F/ x ps
vtoclf.2  |-  A  e. 
_V
vtoclf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtoclf.4  |-  ph
Assertion
Ref Expression
vtoclf  |-  ps
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem vtoclf
StepHypRef Expression
1 vtoclf.1 . . 3  |-  F/ x ps
2 vtoclf.2 . . . . 5  |-  A  e. 
_V
32isseti 2757 . . . 4  |-  E. x  x  =  A
4 vtoclf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54biimpd 144 . . . 4  |-  ( x  =  A  ->  ( ph  ->  ps ) )
63, 5eximii 1612 . . 3  |-  E. x
( ph  ->  ps )
71, 619.36i 1682 . 2  |-  ( A. x ph  ->  ps )
8 vtoclf.4 . 2  |-  ph
97, 8mpg 1461 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1363   F/wnf 1470    e. wcel 2158   _Vcvv 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751
This theorem is referenced by:  vtocl  2803
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