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Theorem ceqsex 3118
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 487 . . 3  |-  ( ( x  =  A  /\  ph )  ->  ps )
41, 3exlimi 1969 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  ps )
52biimprcd 229 . . . 4  |-  ( ps 
->  ( x  =  A  ->  ph ) )
61, 5alrimi 1929 . . 3  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
7 ceqsex.2 . . . 4  |-  A  e. 
_V
87isseti 3088 . . 3  |-  E. x  x  =  A
9 exintr 1747 . . 3  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x
( x  =  A  /\  ph ) ) )
106, 8, 9mpisyl 22 . 2  |-  ( ps 
->  E. x ( x  =  A  /\  ph ) )
114, 10impbii 191 1  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1436    = wceq 1438   E.wex 1660   F/wnf 1664    e. wcel 1869   _Vcvv 3082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-12 1906  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-v 3084
This theorem is referenced by:  ceqsexv  3119  ceqsex2  3120
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