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Theorem 19.23h 1728
Description: Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
Hypothesis
Ref Expression
19.23h.1  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
19.23h  |-  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) )

Proof of Theorem 19.23h
StepHypRef Expression
1 exim 1562 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
2 19.23h.1 . . . 4  |-  ( ps 
->  A. x ps )
3219.9h 1727 . . 3  |-  ( E. x ps  <->  ps )
41, 3syl6ib 217 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  ps )
)
5 hbe1 1705 . . . 4  |-  ( E. x ph  ->  A. x E. x ph )
65, 2hbim 1725 . . 3  |-  ( ( E. x ph  ->  ps )  ->  A. x
( E. x ph  ->  ps ) )
7 19.8a 1718 . . . 4  |-  ( ph  ->  E. x ph )
87imim1i 54 . . 3  |-  ( ( E. x ph  ->  ps )  ->  ( ph  ->  ps ) )
96, 8alrimih 1552 . 2  |-  ( ( E. x ph  ->  ps )  ->  A. x
( ph  ->  ps )
)
104, 9impbii 180 1  |-  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528
This theorem is referenced by:  exlimih  1729  a12study9  29120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-ex 1529
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