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Theorem exim 1573
Description: Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
Assertion
Ref Expression
exim  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )

Proof of Theorem exim
StepHypRef Expression
1 con3 128 . . . 4  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
21al2imi 1549 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x  -.  ps  ->  A. x  -.  ph ) )
3 alnex 1569 . . 3  |-  ( A. x  -.  ps  <->  -.  E. x ps )
4 alnex 1569 . . 3  |-  ( A. x  -.  ph  <->  -.  E. x ph )
52, 3, 43imtr3g 262 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( -.  E. x ps  ->  -.  E. x ph ) )
65con4d 99 1  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532   E.wex 1537
This theorem is referenced by:  eximi  1574  exbi  1579  eximdh  1586  19.29  1595  19.25  1602  19.30  1603  ax12o10lem14  1648  19.23t  1776  2mo  2191  elex22  2738  elex2  2739  vtoclegft  2793  cla4imgft  2797  ssoprab2  5756  2exim  26743  pm11.71  26762  onfrALTlem2  27101  19.41rg  27106  a9e2nd  27114  elex2VD  27401  elex22VD  27402  onfrALTlem2VD  27452  19.41rgVD  27465  a9e2eqVD  27470  a9e2ndVD  27471  a9e2ndeqVD  27472  a9e2ndALT  27494  a9e2ndeqALT  27495  ax12o10lem14K  27960  ax12o10lem14X  27961
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536
This theorem depends on definitions:  df-bi 179  df-ex 1538
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