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Theorem ceqsalg 2688
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
ceqsalg.1  |-  F/ x ps
ceqsalg.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsalg  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem ceqsalg
StepHypRef Expression
1 elisset 2674 . . 3  |-  ( A  e.  V  ->  E. x  x  =  A )
2 nfa1 1506 . . . 4  |-  F/ x A. x ( x  =  A  ->  ph )
3 ceqsalg.1 . . . 4  |-  F/ x ps
4 ceqsalg.2 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54biimpd 143 . . . . . 6  |-  ( x  =  A  ->  ( ph  ->  ps ) )
65a2i 11 . . . . 5  |-  ( ( x  =  A  ->  ph )  ->  ( x  =  A  ->  ps ) )
76sps 1502 . . . 4  |-  ( A. x ( x  =  A  ->  ph )  -> 
( x  =  A  ->  ps ) )
82, 3, 7exlimd 1561 . . 3  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  ps ) )
91, 8syl5com 29 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  ->  ps ) )
104biimprcd 159 . . 3  |-  ( ps 
->  ( x  =  A  ->  ph ) )
113, 10alrimi 1487 . 2  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
129, 11impbid1 141 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1314    = wceq 1316   F/wnf 1421   E.wex 1453    e. wcel 1465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by:  ceqsal  2689  sbc6g  2906  uniiunlem  3155  sucprcreg  4434  funimass4  5440  ralrnmpo  5853
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