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Theorem 19.33b 1618
Description: The antecedent provides a condition implying the converse of 19.33 1617. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)
Assertion
Ref Expression
19.33b  |-  ( -.  ( E. x ph  /\ 
E. x ps )  ->  ( A. x (
ph  \/  ps )  <->  ( A. x ph  \/  A. x ps ) ) )

Proof of Theorem 19.33b
StepHypRef Expression
1 ianor 475 . . 3  |-  ( -.  ( E. x ph  /\ 
E. x ps )  <->  ( -.  E. x ph  \/  -.  E. x ps ) )
2 alnex 1552 . . . . . 6  |-  ( A. x  -.  ph  <->  -.  E. x ph )
3 pm2.53 363 . . . . . . 7  |-  ( (
ph  \/  ps )  ->  ( -.  ph  ->  ps ) )
43al2imi 1570 . . . . . 6  |-  ( A. x ( ph  \/  ps )  ->  ( A. x  -.  ph  ->  A. x ps ) )
52, 4syl5bir 210 . . . . 5  |-  ( A. x ( ph  \/  ps )  ->  ( -. 
E. x ph  ->  A. x ps ) )
6 olc 374 . . . . 5  |-  ( A. x ps  ->  ( A. x ph  \/  A. x ps ) )
75, 6syl6com 33 . . . 4  |-  ( -. 
E. x ph  ->  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  A. x ps ) ) )
8 19.30 1614 . . . . . . 7  |-  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) )
98orcomd 378 . . . . . 6  |-  ( A. x ( ph  \/  ps )  ->  ( E. x ps  \/  A. x ph ) )
109ord 367 . . . . 5  |-  ( A. x ( ph  \/  ps )  ->  ( -. 
E. x ps  ->  A. x ph ) )
11 orc 375 . . . . 5  |-  ( A. x ph  ->  ( A. x ph  \/  A. x ps ) )
1210, 11syl6com 33 . . . 4  |-  ( -. 
E. x ps  ->  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  A. x ps ) ) )
137, 12jaoi 369 . . 3  |-  ( ( -.  E. x ph  \/  -.  E. x ps )  ->  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  A. x ps ) ) )
141, 13sylbi 188 . 2  |-  ( -.  ( E. x ph  /\ 
E. x ps )  ->  ( A. x (
ph  \/  ps )  ->  ( A. x ph  \/  A. x ps )
) )
15 19.33 1617 . 2  |-  ( ( A. x ph  \/  A. x ps )  ->  A. x ( ph  \/  ps ) )
1614, 15impbid1 195 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 177    \/ wo 358    /\ wa 359   A.wal 1549   E.wex 1550
This theorem is referenced by:  kmlem16  8034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-ex 1551
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