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Theorem 19.31r 1616
Description: One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
Hypothesis
Ref Expression
19.31r.1  |-  F/ x ps
Assertion
Ref Expression
19.31r  |-  ( ( A. x ph  \/  ps )  ->  A. x
( ph  \/  ps ) )

Proof of Theorem 19.31r
StepHypRef Expression
1 19.31r.1 . . 3  |-  F/ x ps
2119.32r 1615 . 2  |-  ( ( ps  \/  A. x ph )  ->  A. x
( ps  \/  ph ) )
3 orcom 682 . 2  |-  ( ( A. x ph  \/  ps )  <->  ( ps  \/  A. x ph ) )
4 orcom 682 . . 3  |-  ( (
ph  \/  ps )  <->  ( ps  \/  ph )
)
54albii 1404 . 2  |-  ( A. x ( ph  \/  ps )  <->  A. x ( ps  \/  ph ) )
62, 3, 53imtr4i 199 1  |-  ( ( A. x ph  \/  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: (None)
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