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Theorem exim 1592
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 hba1 1533 . 2  |-  ( A. x ( ph  ->  ps )  ->  A. x A. x ( ph  ->  ps ) )
2 hbe1 1488 . 2  |-  ( E. x ps  ->  A. x E. x ps )
3 19.8a 1583 . . . 4  |-  ( ps 
->  E. x ps )
43imim2i 12 . . 3  |-  ( (
ph  ->  ps )  -> 
( ph  ->  E. x ps ) )
54sps 1530 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  E. x ps )
)
61, 2, 5exlimdh 1589 1  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346   E.wex 1485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  eximi  1593  exbi  1597  eximdh  1604  19.29  1613  19.25  1619  alexim  1638  19.23t  1670  spimt  1729  equvini  1751  nfexd  1754  ax10oe  1790  sbcof2  1803  spsbim  1836  nf5-1  2017  mor  2061  rexim  2564  elex22  2745  elex2  2746  vtoclegft  2802  spcimgft  2806  spcimegft  2808  spc2gv  2821  spc3gv  2823  ssoprab2  5906  bj-inf2vnlem1  13927
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