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Theorem ceqsalg 2669
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 2655 . . 3  |-  ( A  e.  V  ->  E. x  x  =  A )
2 nfa1 1489 . . . 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 1485 . . . 4  |-  ( A. x ( x  =  A  ->  ph )  -> 
( x  =  A  ->  ps ) )
82, 3, 7exlimd 1544 . . 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 1470 . 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 1297    = wceq 1299   F/wnf 1404   E.wex 1436    e. wcel 1448
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 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-v 2643
This theorem is referenced by:  ceqsal  2670  sbc6g  2886  uniiunlem  3132  sucprcreg  4402  funimass4  5404  ralrnmpo  5817
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