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Theorem speiv 1940
Description: Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)
Hypotheses
Ref Expression
speiv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
speiv.2  |-  ps
Assertion
Ref Expression
speiv  |-  E. x ph
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)

Proof of Theorem speiv
StepHypRef Expression
1 speiv.2 . 2  |-  ps
2 speiv.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32biimprd 214 . . 3  |-  ( x  =  y  ->  ( ps  ->  ph ) )
43spimev 1939 . 2  |-  ( ps 
->  E. x ph )
51, 4ax-mp 8 1  |-  E. x ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1528
This theorem is referenced by:  elirrv  7311
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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