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Theorem ceqsal 2649
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
ceqsal.1  |-  F/ x ps
ceqsal.2  |-  A  e. 
_V
ceqsal.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsal  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem ceqsal
StepHypRef Expression
1 ceqsal.2 . 2  |-  A  e. 
_V
2 ceqsal.1 . . 3  |-  F/ x ps
3 ceqsal.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
42, 3ceqsalg 2648 . 2  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
51, 4ax-mp 7 1  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1288    = wceq 1290   F/wnf 1395    e. wcel 1439   _Vcvv 2620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-v 2622
This theorem is referenced by:  ceqsalv  2650
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