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Theorem 3jao 1287
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 972 . 2  |-  ( (
ph  \/  ch  \/  th )  <->  ( ( ph  \/  ch )  \/  th ) )
2 jao 510 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ( ch  ->  ps )  ->  ( ( ph  \/  ch )  ->  ps ) ) )
3 jao 510 . . . 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 1188 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th 
->  ps ) )  -> 
( ( ( ph  \/  ch )  \/  th )  ->  ps ) )
61, 5syl5bi 217 1  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th 
->  ps ) )  -> 
( ( ph  \/  ch  \/  th )  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    \/ w3o 970    /\ w3a 971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973
This theorem is referenced by:  3jaob  1288  3jaoi  1289  3jaod  1290
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