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Theorem ceqsexg 3145
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
Hypotheses
Ref Expression
ceqsexg.1  |-  F/ x ps
ceqsexg.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsexg  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem ceqsexg
StepHypRef Expression
1 nfe1 1894 . . 3  |-  F/ x E. x ( x  =  A  /\  ph )
2 ceqsexg.1 . . 3  |-  F/ x ps
31, 2nfbi 1994 . 2  |-  F/ x
( E. x ( x  =  A  /\  ph )  <->  ps )
4 ceqex 3144 . . 3  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
5 ceqsexg.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
64, 5bibi12d 322 . 2  |-  ( x  =  A  ->  (
( ph  <->  ph )  <->  ( E. x ( x  =  A  /\  ph )  <->  ps ) ) )
7 biid 239 . 2  |-  ( ph  <->  ph )
83, 6, 7vtoclg1f 3081 1  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657   F/wnf 1661    e. wcel 1872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3024
This theorem is referenced by:  ceqsexgv  3146
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