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Theorem spimed 1917
Description: Deduction version of spime 1916. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
spimed.1  |-  ( ch 
->  F/ x ph )
spimed.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spimed  |-  ( ch 
->  ( ph  ->  E. x ps ) )

Proof of Theorem spimed
StepHypRef Expression
1 spimed.1 . 2  |-  ( ch 
->  F/ x ph )
2 nfnf1 1757 . . . . 5  |-  F/ x F/ x ph
3 id 19 . . . . 5  |-  ( F/ x ph  ->  F/ x ph )
42, 3nfan1 1822 . . . 4  |-  F/ x
( F/ x ph  /\ 
ph )
5 spimed.2 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  ps ) )
65adantld 453 . . . 4  |-  ( x  =  y  ->  (
( F/ x ph  /\ 
ph )  ->  ps ) )
74, 6spime 1916 . . 3  |-  ( ( F/ x ph  /\  ph )  ->  E. x ps )
87ex 423 . 2  |-  ( F/ x ph  ->  ( ph  ->  E. x ps )
)
91, 8syl 15 1  |-  ( ch 
->  ( ph  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528   F/wnf 1531
This theorem is referenced by:  equvini  1927
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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