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Theorem ceqsalv 2814
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
Hypotheses
Ref Expression
ceqsalv.1  |-  A  e. 
_V
ceqsalv.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsalv  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsalv
StepHypRef Expression
1 nfv 1605 . 2  |-  F/ x ps
2 ceqsalv.1 . 2  |-  A  e. 
_V
3 ceqsalv.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3ceqsal 2813 1  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788
This theorem is referenced by:  clel2  2904  clel4  2907  reu8  2961  frsn  4760  raliunxp  4825  fv3  5541  funimass4  5573  marypha2lem3  7190  kmlem12  7787  fpwwe2lem12  8263  vdwmc2  13026  itg2leub  19089  nmoubi  21350  choc0  21905  nmopub  22488  nmfnleub  22505  heibor1lem  26533  ralxpxfr2d  26760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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