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Theorem spimed 1961
Description: Deduction version of spime 1963. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
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 . . 3  |-  ( ch 
->  F/ x ph )
21nfrd 1780 . 2  |-  ( ch 
->  ( ph  ->  A. x ph ) )
3 a9e 1953 . . . 4  |-  E. x  x  =  y
4 spimed.2 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
53, 4eximii 1588 . . 3  |-  E. x
( ph  ->  ps )
6519.35i 1612 . 2  |-  ( A. x ph  ->  E. x ps )
72, 6syl6 32 1  |-  ( ch 
->  ( ph  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550   E.wex 1551   F/wnf 1554
This theorem is referenced by:  spime  1963  equvini  2084  equviniOLD  2085
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-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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