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Theorem ceqsexv2d 25882
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
Hypotheses
Ref Expression
ceqsexv2d.1  |-  A  e. 
_V
ceqsexv2d.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsexv2d.3  |-  ps
Assertion
Ref Expression
ceqsexv2d  |-  E. x ph
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsexv2d
StepHypRef Expression
1 ceqsexv2d.3 . 2  |-  ps
2 ceqsexv2d.1 . . . 4  |-  A  e. 
_V
3 ceqsexv2d.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
42, 3ceqsexv 3009 . . 3  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
54biimpri 206 . 2  |-  ( ps 
->  E. x ( x  =  A  /\  ph ) )
6 simpr 461 . . 3  |-  ( ( x  =  A  /\  ph )  ->  ph )
76eximi 1625 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  E. x ph )
81, 5, 7mp2b 10 1  |-  E. x ph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-12 1792  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-v 2974
This theorem is referenced by: (None)
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