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Theorem 3jaob 1244
Description: Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
3jaob  |-  ( ( ( ph  \/  ch  \/  th )  ->  ps ) 
<->  ( ( ph  ->  ps )  /\  ( ch 
->  ps )  /\  ( th  ->  ps ) ) )

Proof of Theorem 3jaob
StepHypRef Expression
1 3mix1 1124 . . . 4  |-  ( ph  ->  ( ph  \/  ch  \/  th ) )
21imim1i 54 . . 3  |-  ( ( ( ph  \/  ch  \/  th )  ->  ps )  ->  ( ph  ->  ps ) )
3 3mix2 1125 . . . 4  |-  ( ch 
->  ( ph  \/  ch  \/  th ) )
43imim1i 54 . . 3  |-  ( ( ( ph  \/  ch  \/  th )  ->  ps )  ->  ( ch  ->  ps ) )
5 3mix3 1126 . . . 4  |-  ( th 
->  ( ph  \/  ch  \/  th ) )
65imim1i 54 . . 3  |-  ( ( ( ph  \/  ch  \/  th )  ->  ps )  ->  ( th  ->  ps ) )
72, 4, 63jca 1132 . 2  |-  ( ( ( ph  \/  ch  \/  th )  ->  ps )  ->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th  ->  ps )
) )
8 3jao 1243 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th 
->  ps ) )  -> 
( ( ph  \/  ch  \/  th )  ->  ps ) )
97, 8impbii 180 1  |-  ( ( ( ph  \/  ch  \/  th )  ->  ps ) 
<->  ( ( ph  ->  ps )  /\  ( ch 
->  ps )  /\  ( th  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ w3o 933    /\ w3a 934
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 177  df-or 359  df-an 360  df-3or 935  df-3an 936
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