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Theorem exbi 1538
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 bi1 116 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
21alimi 1387 . . 3  |-  ( A. x ( ph  <->  ps )  ->  A. x ( ph  ->  ps ) )
3 exim 1533 . . 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 bi2 128 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
65alimi 1387 . . 3  |-  ( A. x ( ph  <->  ps )  ->  A. x ( ps 
->  ph ) )
7 exim 1533 . . 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 127 1  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ph  <->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1285   E.wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  exbii  1539  exbidh  1548  exintrbi  1567  19.19  1599  rexrnmpt  5399
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