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Theorem spime 1929
Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
spime.1  |-  F/ x ph
spime.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spime  |-  ( ph  ->  E. x ps )

Proof of Theorem spime
StepHypRef Expression
1 spime.1 . . . . 5  |-  F/ x ph
21nfn 1777 . . . 4  |-  F/ x  -.  ph
3 spime.2 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  ps ) )
43con3d 125 . . . 4  |-  ( x  =  y  ->  ( -.  ps  ->  -.  ph )
)
52, 4spim 1928 . . 3  |-  ( A. x  -.  ps  ->  -.  ph )
65con2i 112 . 2  |-  ( ph  ->  -.  A. x  -.  ps )
7 df-ex 1532 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
86, 7sylibr 203 1  |-  ( ph  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   E.wex 1531   F/wnf 1534
This theorem is referenced by:  spimed  1930  spimev  1952  exnel  24230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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