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Theorem ceqsalv 2927
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 1626 . 2  |-  F/ x ps
2 ceqsalv.1 . 2  |-  A  e. 
_V
3 ceqsalv.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3ceqsal 2926 1  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   _Vcvv 2901
This theorem is referenced by:  clel2  3017  clel4  3020  reu8  3075  frsn  4890  raliunxp  4956  fv3  5686  funimass4  5718  marypha2lem3  7379  kmlem12  7976  fpwwe2lem12  8451  vdwmc2  13276  itg2leub  19495  nmoubi  22123  choc0  22678  nmopub  23261  nmfnleub  23278  heibor1lem  26211  ralxpxfr2d  26434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-v 2903
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