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Theorem vtoclf 3160
Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2014. (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 3115 . . . 4  |-  E. x  x  =  A
4 vtoclf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54biimpd 207 . . . 4  |-  ( x  =  A  ->  ( ph  ->  ps ) )
63, 5eximii 1659 . . 3  |-  E. x
( ph  ->  ps )
71, 619.36i 1966 . 2  |-  ( A. x ph  ->  ps )
8 vtoclf.4 . 2  |-  ph
97, 8mpg 1621 1  |-  ps
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395   F/wnf 1617    e. wcel 1819   _Vcvv 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111
This theorem is referenced by:  vtocl  3161  summolem2a  13549  prodmolem2a  13753  monotuz  31081  oddcomabszz  31084  binomcxplemnotnn0  31465  limclner  31860  dvnmptdivc  31938  dvnmul  31943
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