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

Step	Hyp	Ref	Expression
1		3mix1 1124	. . . 4 |- (ph -> (ph \/ ch \/ th))
2	1	imim1i 54	. . 3 |- (((ph \/ ch \/ th) -> ps) -> (ph -> ps))
3		3mix2 1125	. . . 4 |- (ch -> (ph \/ ch \/ th))
4	3	imim1i 54	. . 3 |- (((ph \/ ch \/ th) -> ps) -> (ch -> ps))
5		3mix3 1126	. . . 4 |- (th -> (ph \/ ch \/ th))
6	5	imim1i 54	. . 3 |- (((ph \/ ch \/ th) -> ps) -> (th -> ps))
7	2, 4, 6	3jca 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))
9	7, 8	impbii 180	1 |- (((ph \/ ch \/ th) -> ps) <-> ((ph -> ps) /\ (ch -> ps) /\ (th -> ps)))

Syntax hints:   -> wi 4   <-> wb 176   \/ 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: (None)