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Theorem 19.33b2 1591
Description: The antecedent provides a condition implying the converse of 19.33 1443. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1592 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 700 . . . . 5  |-  ( ( -.  E. x ph  \/  -.  E. x ps )  <->  ( -.  E. x ps  \/  -.  E. x ph ) )
2 alnex 1458 . . . . . 6  |-  ( A. x  -.  ps  <->  -.  E. x ps )
3 alnex 1458 . . . . . 6  |-  ( A. x  -.  ph  <->  -.  E. x ph )
42, 3orbi12i 736 . . . . 5  |-  ( ( A. x  -.  ps  \/  A. x  -.  ph ) 
<->  ( -.  E. x ps  \/  -.  E. x ph ) )
51, 4bitr4i 186 . . . 4  |-  ( ( -.  E. x ph  \/  -.  E. x ps )  <->  ( A. x  -.  ps  \/  A. x  -.  ph ) )
6 pm2.53 694 . . . . . . 7  |-  ( ( ps  \/  ph )  ->  ( -.  ps  ->  ph ) )
76orcoms 702 . . . . . 6  |-  ( (
ph  \/  ps )  ->  ( -.  ps  ->  ph ) )
87al2imi 1417 . . . . 5  |-  ( A. x ( ph  \/  ps )  ->  ( A. x  -.  ps  ->  A. x ph ) )
9 pm2.53 694 . . . . . 6  |-  ( (
ph  \/  ps )  ->  ( -.  ph  ->  ps ) )
109al2imi 1417 . . . . 5  |-  ( A. x ( ph  \/  ps )  ->  ( A. x  -.  ph  ->  A. x ps ) )
118, 10orim12d 758 . . . 4  |-  ( A. x ( ph  \/  ps )  ->  ( ( A. x  -.  ps  \/  A. x  -.  ph )  ->  ( A. x ph  \/  A. x ps ) ) )
125, 11syl5bi 151 . . 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 1443 . 2  |-  ( ( A. x ph  \/  A. x ps )  ->  A. x ( ph  \/  ps ) )
1513, 14impbid1 141 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 104    \/ wo 680   A.wal 1312   E.wex 1451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-gen 1408  ax-ie2 1453
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320
This theorem is referenced by:  19.33bdc  1592
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