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Theorem 19.30dc 1534
Description: Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)
Assertion
Ref Expression
19.30dc  |-  (DECID  E. x ps  ->  ( A. x
( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) ) )

Proof of Theorem 19.30dc
StepHypRef Expression
1 df-dc 754 . 2  |-  (DECID  E. x ps 
<->  ( E. x ps  \/  -.  E. x ps ) )
2 olc 642 . . . 4  |-  ( E. x ps  ->  ( A. x ph  \/  E. x ps ) )
32a1d 22 . . 3  |-  ( E. x ps  ->  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) ) )
4 alnex 1404 . . . . 5  |-  ( A. x  -.  ps  <->  -.  E. x ps )
5 orel2 655 . . . . . 6  |-  ( -. 
ps  ->  ( ( ph  \/  ps )  ->  ph )
)
65al2imi 1363 . . . . 5  |-  ( A. x  -.  ps  ->  ( A. x ( ph  \/  ps )  ->  A. x ph ) )
74, 6sylbir 129 . . . 4  |-  ( -. 
E. x ps  ->  ( A. x ( ph  \/  ps )  ->  A. x ph ) )
8 orc 643 . . . 4  |-  ( A. x ph  ->  ( A. x ph  \/  E. x ps ) )
97, 8syl6 33 . . 3  |-  ( -. 
E. x ps  ->  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) ) )
103, 9jaoi 646 . 2  |-  ( ( E. x ps  \/  -.  E. x ps )  ->  ( A. x (
ph  \/  ps )  ->  ( A. x ph  \/  E. x ps )
) )
111, 10sylbi 118 1  |-  (DECID  E. x ps  ->  ( A. x
( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 639  DECID wdc 753   A.wal 1257   E.wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie2 1399
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-fal 1265
This theorem is referenced by: (None)
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