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Theorem ceqsalv 2827
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 1609 . 2  |-  F/ x ps
2 ceqsalv.1 . 2  |-  A  e. 
_V
3 ceqsalv.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3ceqsal 2826 1  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  clel2  2917  clel4  2920  reu8  2974  frsn  4776  raliunxp  4841  fv3  5557  funimass4  5589  marypha2lem3  7206  kmlem12  7803  fpwwe2lem12  8279  vdwmc2  13042  itg2leub  19105  nmoubi  21366  choc0  21921  nmopub  22504  nmfnleub  22521  heibor1lem  26636  ralxpxfr2d  26863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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