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Theorem a4ime 1869
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
a4ime.1  |-  F/ x ph
a4ime.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
a4ime  |-  ( ph  ->  E. x ps )

Proof of Theorem a4ime
StepHypRef Expression
1 a4ime.1 . . . . 5  |-  F/ x ph
21nfn 1731 . . . 4  |-  F/ x  -.  ph
3 a4ime.2 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  ps ) )
43con3d 127 . . . 4  |-  ( x  =  y  ->  ( -.  ps  ->  -.  ph )
)
52, 4a4im 1868 . . 3  |-  ( A. x  -.  ps  ->  -.  ph )
65con2i 114 . 2  |-  ( ph  ->  -.  A. x  -.  ps )
7 df-ex 1538 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
86, 7sylibr 205 1  |-  ( ph  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532   E.wex 1537   F/wnf 1539    = wceq 1619
This theorem is referenced by:  a4imed  1870  a4imev  1998  exnel  23528
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540
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