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Theorem ceqsexg 3231
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 1841 . . 3  |-  F/ x E. x ( x  =  A  /\  ph )
2 ceqsexg.1 . . 3  |-  F/ x ps
31, 2nfbi 1935 . 2  |-  F/ x
( E. x ( x  =  A  /\  ph )  <->  ps )
4 ceqex 3230 . . 3  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
5 ceqsexg.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
64, 5bibi12d 321 . 2  |-  ( x  =  A  ->  (
( ph  <->  ph )  <->  ( E. x ( x  =  A  /\  ph )  <->  ps ) ) )
7 biid 236 . 2  |-  ( ph  <->  ph )
83, 6, 7vtoclg1f 3166 1  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   E.wex 1613   F/wnf 1617    e. wcel 1819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111
This theorem is referenced by:  ceqsexgv  3232
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