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Theorem ceqsexg 2899
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 nfcv 2419 . 2  |-  F/_ x A
2 nfe1 1706 . . 3  |-  F/ x E. x ( x  =  A  /\  ph )
3 ceqsexg.1 . . 3  |-  F/ x ps
42, 3nfbi 1772 . 2  |-  F/ x
( E. x ( x  =  A  /\  ph )  <->  ps )
5 ceqex 2898 . . 3  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
6 ceqsexg.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6bibi12d 312 . 2  |-  ( x  =  A  ->  (
( ph  <->  ph )  <->  ( E. x ( x  =  A  /\  ph )  <->  ps ) ) )
8 biid 227 . 2  |-  ( ph  <->  ph )
91, 4, 7, 8vtoclgf 2842 1  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528   F/wnf 1531    = wceq 1623    e. wcel 1684
This theorem is referenced by:  ceqsexgv  2900
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-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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