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Theorem ceqsalg 2812
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 2798 . . 3  |-  ( A  e.  V  ->  E. x  x  =  A )
2 nfa1 1756 . . . 4  |-  F/ x A. x ( x  =  A  ->  ph )
3 ceqsalg.1 . . . 4  |-  F/ x ps
4 ceqsalg.2 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54biimpd 198 . . . . . 6  |-  ( x  =  A  ->  ( ph  ->  ps ) )
65a2i 12 . . . . 5  |-  ( ( x  =  A  ->  ph )  ->  ( x  =  A  ->  ps ) )
76sps 1739 . . . 4  |-  ( A. x ( x  =  A  ->  ph )  -> 
( x  =  A  ->  ps ) )
82, 3, 7exlimd 1803 . . 3  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  ps ) )
91, 8syl5com 26 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  ->  ps ) )
104biimprcd 216 . . 3  |-  ( ps 
->  ( x  =  A  ->  ph ) )
113, 10alrimi 1745 . 2  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
129, 11impbid1 194 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528   F/wnf 1531    = wceq 1623    e. wcel 1684
This theorem is referenced by:  ceqsal  2813  sbc6g  3016  uniiunlem  3260  ralrnmpt2  5958  sucprcreg  7313  fimaxre3  9703  pmapglbx  29958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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