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Theorem ceqsalv 2974
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 1629 . 2  |-  F/ x ps
2 ceqsalv.1 . 2  |-  A  e. 
_V
3 ceqsalv.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3ceqsal 2973 1  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948
This theorem is referenced by:  clel2  3064  clel4  3067  reu8  3122  frsn  4940  raliunxp  5006  fv3  5736  funimass4  5769  marypha2lem3  7434  kmlem12  8033  fpwwe2lem12  8508  vdwmc2  13339  itg2leub  19618  nmoubi  22265  choc0  22820  nmopub  23403  nmfnleub  23420  heibor1lem  26509  ralxpxfr2d  26732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950
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