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Theorem ceqsex 2992
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
ceqsex.1  |-  F/ x ps
ceqsex.2  |-  A  e. 
_V
ceqsex.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsex  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem ceqsex
StepHypRef Expression
1 ceqsex.1 . . 3  |-  F/ x ps
2 ceqsex.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32biimpa 472 . . 3  |-  ( ( x  =  A  /\  ph )  ->  ps )
41, 3exlimi 1822 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  ps )
52biimprcd 218 . . . 4  |-  ( ps 
->  ( x  =  A  ->  ph ) )
61, 5alrimi 1782 . . 3  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
7 ceqsex.2 . . . 4  |-  A  e. 
_V
87isseti 2964 . . 3  |-  E. x  x  =  A
9 exintr 1625 . . 3  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x
( x  =  A  /\  ph ) ) )
106, 8, 9ee10 1386 . 2  |-  ( ps 
->  E. x ( x  =  A  /\  ph ) )
114, 10impbii 182 1  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551   F/wnf 1554    = wceq 1653    e. wcel 1726   _Vcvv 2958
This theorem is referenced by:  ceqsexv  2993  ceqsex2  2994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
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