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Theorem exbi 1604
Description: Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
exbi  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ph  <->  E. x ps ) )

Proof of Theorem exbi
StepHypRef Expression
1 biimp 118 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
21alimi 1455 . . 3  |-  ( A. x ( ph  <->  ps )  ->  A. x ( ph  ->  ps ) )
3 exim 1599 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
42, 3syl 14 . 2  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ph  ->  E. x ps )
)
5 biimpr 130 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
65alimi 1455 . . 3  |-  ( A. x ( ph  <->  ps )  ->  A. x ( ps 
->  ph ) )
7 exim 1599 . . 3  |-  ( A. x ( ps  ->  ph )  ->  ( E. x ps  ->  E. x ph ) )
86, 7syl 14 . 2  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ps 
->  E. x ph )
)
94, 8impbid 129 1  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ph  <->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351   E.wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  exbii  1605  exbidh  1614  exintrbi  1633  19.19  1666  rexrnmpt  5655
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