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Theorem 19.33b2 1565
Description: The antecedent provides a condition implying the converse of 19.33 1418. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1566 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.33b2  |-  ( ( -.  E. x ph  \/  -.  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph  \/  A. x ps ) ) )

Proof of Theorem 19.33b2
StepHypRef Expression
1 orcom 682 . . . . 5  |-  ( ( -.  E. x ph  \/  -.  E. x ps )  <->  ( -.  E. x ps  \/  -.  E. x ph ) )
2 alnex 1433 . . . . . 6  |-  ( A. x  -.  ps  <->  -.  E. x ps )
3 alnex 1433 . . . . . 6  |-  ( A. x  -.  ph  <->  -.  E. x ph )
42, 3orbi12i 716 . . . . 5  |-  ( ( A. x  -.  ps  \/  A. x  -.  ph ) 
<->  ( -.  E. x ps  \/  -.  E. x ph ) )
51, 4bitr4i 185 . . . 4  |-  ( ( -.  E. x ph  \/  -.  E. x ps )  <->  ( A. x  -.  ps  \/  A. x  -.  ph ) )
6 pm2.53 676 . . . . . . 7  |-  ( ( ps  \/  ph )  ->  ( -.  ps  ->  ph ) )
76orcoms 684 . . . . . 6  |-  ( (
ph  \/  ps )  ->  ( -.  ps  ->  ph ) )
87al2imi 1392 . . . . 5  |-  ( A. x ( ph  \/  ps )  ->  ( A. x  -.  ps  ->  A. x ph ) )
9 pm2.53 676 . . . . . 6  |-  ( (
ph  \/  ps )  ->  ( -.  ph  ->  ps ) )
109al2imi 1392 . . . . 5  |-  ( A. x ( ph  \/  ps )  ->  ( A. x  -.  ph  ->  A. x ps ) )
118, 10orim12d 735 . . . 4  |-  ( A. x ( ph  \/  ps )  ->  ( ( A. x  -.  ps  \/  A. x  -.  ph )  ->  ( A. x ph  \/  A. x ps ) ) )
125, 11syl5bi 150 . . 3  |-  ( A. x ( ph  \/  ps )  ->  ( ( -.  E. x ph  \/  -.  E. x ps )  ->  ( A. x ph  \/  A. x ps ) ) )
1312com12 30 . 2  |-  ( ( -.  E. x ph  \/  -.  E. x ps )  ->  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  A. x ps ) ) )
14 19.33 1418 . 2  |-  ( ( A. x ph  \/  A. x ps )  ->  A. x ( ph  \/  ps ) )
1513, 14impbid1 140 1  |-  ( ( -.  E. x ph  \/  -.  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph  \/  A. x ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 664   A.wal 1287   E.wex 1426
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie2 1428
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295
This theorem is referenced by:  19.33bdc  1566
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