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Theorem 19.32r 1615
Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if  ph is decidable, as seen at 19.32dc 1614. (Contributed by Jim Kingdon, 28-Jul-2018.)
Hypothesis
Ref Expression
19.32r.1  |-  F/ x ph
Assertion
Ref Expression
19.32r  |-  ( (
ph  \/  A. x ps )  ->  A. x
( ph  \/  ps ) )

Proof of Theorem 19.32r
StepHypRef Expression
1 19.32r.1 . . 3  |-  F/ x ph
2 orc 668 . . 3  |-  ( ph  ->  ( ph  \/  ps ) )
31, 2alrimi 1460 . 2  |-  ( ph  ->  A. x ( ph  \/  ps ) )
4 olc 667 . . 3  |-  ( ps 
->  ( ph  \/  ps ) )
54alimi 1389 . 2  |-  ( A. x ps  ->  A. x
( ph  \/  ps ) )
63, 5jaoi 671 1  |-  ( (
ph  \/  A. x ps )  ->  A. x
( ph  \/  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 664   A.wal 1287   F/wnf 1394
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-io 665  ax-5 1381  ax-gen 1383  ax-4 1445
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  19.31r  1616
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