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Theorem 19.30 1593
Description: Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
19.30  |-  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) )

Proof of Theorem 19.30
StepHypRef Expression
1 exnal 1563 . . 3  |-  ( E. x  -.  ph  <->  -.  A. x ph )
2 exim 1564 . . 3  |-  ( A. x ( -.  ph  ->  ps )  ->  ( E. x  -.  ph  ->  E. x ps ) )
31, 2syl5bir 209 . 2  |-  ( A. x ( -.  ph  ->  ps )  ->  ( -.  A. x ph  ->  E. x ps ) )
4 df-or 359 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
54albii 1555 . 2  |-  ( A. x ( ph  \/  ps )  <->  A. x ( -. 
ph  ->  ps ) )
6 df-or 359 . 2  |-  ( ( A. x ph  \/  E. x ps )  <->  ( -.  A. x ph  ->  E. x ps ) )
73, 5, 63imtr4i 257 1  |-  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357   A.wal 1529   E.wex 1530
This theorem is referenced by:  19.33b  1597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546
This theorem depends on definitions:  df-bi 177  df-or 359  df-ex 1531
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