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Theorem speiv 1818
Description: Inference from existential specialization, using implicit substitition. (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 157 . . 3  |-  ( x  =  y  ->  ( ps  ->  ph ) )
43spimev 1817 . 2  |-  ( ps 
->  E. x ph )
51, 4ax-mp 5 1  |-  E. x ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   E.wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422
This theorem is referenced by: (None)
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