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Theorem pm4.78 566
Description: Implication distributes over disjunction. Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
Assertion
Ref Expression
pm4.78  |-  ( ( ( ph  ->  ps )  \/  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  \/  ch ) ) )

Proof of Theorem pm4.78
StepHypRef Expression
1 orordi 517 . 2  |-  ( ( -.  ph  \/  ( ps  \/  ch ) )  <-> 
( ( -.  ph  \/  ps )  \/  ( -.  ph  \/  ch )
) )
2 imor 402 . 2  |-  ( (
ph  ->  ( ps  \/  ch ) )  <->  ( -.  ph  \/  ( ps  \/  ch ) ) )
3 imor 402 . . 3  |-  ( (
ph  ->  ps )  <->  ( -.  ph  \/  ps ) )
4 imor 402 . . 3  |-  ( (
ph  ->  ch )  <->  ( -.  ph  \/  ch ) )
53, 4orbi12i 508 . 2  |-  ( ( ( ph  ->  ps )  \/  ( ph  ->  ch ) )  <->  ( ( -.  ph  \/  ps )  \/  ( -.  ph  \/  ch ) ) )
61, 2, 53bitr4ri 270 1  |-  ( ( ( ph  ->  ps )  \/  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  \/  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-or 360
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