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Theorem vtoclf 2666
Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1684. (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 2621 . . . 4  |-  E. x  x  =  A
4 vtoclf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54biimpd 142 . . . 4  |-  ( x  =  A  ->  ( ph  ->  ps ) )
63, 5eximii 1536 . . 3  |-  E. x
( ph  ->  ps )
71, 619.36i 1605 . 2  |-  ( A. x ph  ->  ps )
8 vtoclf.4 . 2  |-  ph
97, 8mpg 1383 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1287   F/wnf 1392    e. wcel 1436   _Vcvv 2615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-v 2617
This theorem is referenced by:  vtocl  2667
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