Theorem 3jao 1243

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

Step	Hyp	Ref	Expression
1		df-3or 935	. 2 |- ((ph \/ ch \/ th) <-> ((ph \/ ch) \/ th))
2		jao 498	. . . 4 |- ((ph -> ps) -> ((ch -> ps) -> ((ph \/ ch) -> ps)))
3		jao 498	. . . 4 |- (((ph \/ ch) -> ps) -> ((th -> ps) -> (((ph \/ ch) \/ th) -> ps)))
4	2, 3	syl6 29	. . 3 |- ((ph -> ps) -> ((ch -> ps) -> ((th -> ps) -> (((ph \/ ch) \/ th) -> ps))))
5	4	3imp 1145	. 2 |- (((ph -> ps) /\ (ch -> ps) /\ (th -> ps)) -> (((ph \/ ch) \/ th) -> ps))
6	1, 5	syl5bi 208	1 |- (((ph -> ps) /\ (ch -> ps) /\ (th -> ps)) -> ((ph \/ ch \/ th) -> ps))

Syntax hints:   -> wi 4   \/ wo 357   \/ w3o 933   /\ w3a 934

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 177  df-or 359  df-an 360  df-3or 935  df-3an 936

This theorem is referenced by:  3jaob  1244  3jaoi  1245  3jaod  1246