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Theorem 19.25 1595
Description: Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
19.25  |-  ( A. y E. x ( ph  ->  ps )  ->  ( E. y A. x ph  ->  E. y E. x ps ) )

Proof of Theorem 19.25
StepHypRef Expression
1 19.35 1592 . . . 4  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
21biimpi 188 . . 3  |-  ( E. x ( ph  ->  ps )  ->  ( A. x ph  ->  E. x ps ) )
32alimi 1551 . 2  |-  ( A. y E. x ( ph  ->  ps )  ->  A. y
( A. x ph  ->  E. x ps )
)
4 exim 1567 . 2  |-  ( A. y ( A. x ph  ->  E. x ps )  ->  ( E. y A. x ph  ->  E. y E. x ps ) )
53, 4syl 17 1  |-  ( A. y E. x ( ph  ->  ps )  ->  ( E. y A. x ph  ->  E. y E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532   E.wex 1533
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1534
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