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Theorem exim 1578
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 1520 . 2  |-  ( A. x ( ph  ->  ps )  ->  A. x A. x ( ph  ->  ps ) )
2 hbe1 1471 . 2  |-  ( E. x ps  ->  A. x E. x ps )
3 19.8a 1569 . . . 4  |-  ( ps 
->  E. x ps )
43imim2i 12 . . 3  |-  ( (
ph  ->  ps )  -> 
( ph  ->  E. x ps ) )
54sps 1517 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  E. x ps )
)
61, 2, 5exlimdh 1575 1  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1329   E.wex 1468
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  eximi  1579  exbi  1583  eximdh  1590  19.29  1599  19.25  1605  alexim  1624  19.23t  1655  spimt  1714  equvini  1731  nfexd  1734  ax10oe  1769  sbcof2  1782  spsbim  1815  mor  2041  rexim  2526  elex22  2701  elex2  2702  vtoclegft  2758  spcimgft  2762  spcimegft  2764  spc2gv  2776  spc3gv  2778  ssoprab2  5827  bj-inf2vnlem1  13182
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