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Theorem dedlema 915
Description: Lemma for iftrue 3398. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
Assertion
Ref Expression
dedlema  |-  ( ph  ->  ( ps  <->  ( ( ps  /\  ph )  \/  ( ch  /\  -.  ph ) ) ) )

Proof of Theorem dedlema
StepHypRef Expression
1 orc 668 . . 3  |-  ( ( ps  /\  ph )  ->  ( ( ps  /\  ph )  \/  ( ch 
/\  -.  ph ) ) )
21expcom 114 . 2  |-  ( ph  ->  ( ps  ->  (
( ps  /\  ph )  \/  ( ch  /\ 
-.  ph ) ) ) )
3 simpl 107 . . . 4  |-  ( ( ps  /\  ph )  ->  ps )
43a1i 9 . . 3  |-  ( ph  ->  ( ( ps  /\  ph )  ->  ps )
)
5 pm2.24 586 . . . 4  |-  ( ph  ->  ( -.  ph  ->  ps ) )
65adantld 272 . . 3  |-  ( ph  ->  ( ( ch  /\  -.  ph )  ->  ps ) )
74, 6jaod 672 . 2  |-  ( ph  ->  ( ( ( ps 
/\  ph )  \/  ( ch  /\  -.  ph )
)  ->  ps )
)
82, 7impbid 127 1  |-  ( ph  ->  ( ps  <->  ( ( ps  /\  ph )  \/  ( ch  /\  -.  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664
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-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  iftrue  3398
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