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Theorem 3jao 1235
Description: Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
Assertion
Ref Expression
3jao  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th 
->  ps ) )  -> 
( ( ph  \/  ch  \/  th )  ->  ps ) )

Proof of Theorem 3jao
StepHypRef Expression
1 df-3or 923 . 2  |-  ( (
ph  \/  ch  \/  th )  <->  ( ( ph  \/  ch )  \/  th ) )
2 jao 705 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ( ch  ->  ps )  ->  ( ( ph  \/  ch )  ->  ps ) ) )
3 jao 705 . . . 4  |-  ( ( ( ph  \/  ch )  ->  ps )  -> 
( ( th  ->  ps )  ->  ( (
( ph  \/  ch )  \/  th )  ->  ps ) ) )
42, 3syl6 33 . . 3  |-  ( (
ph  ->  ps )  -> 
( ( ch  ->  ps )  ->  ( ( th  ->  ps )  -> 
( ( ( ph  \/  ch )  \/  th )  ->  ps ) ) ) )
543imp 1135 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th 
->  ps ) )  -> 
( ( ( ph  \/  ch )  \/  th )  ->  ps ) )
61, 5syl5bi 150 1  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th 
->  ps ) )  -> 
( ( ph  \/  ch  \/  th )  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    \/ w3o 921    /\ w3a 922
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 663
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924
This theorem is referenced by:  3jaob  1236  3jaoi  1237  3jaod  1238
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