Theorem 3jaob 1390

Description: Disjunction of three 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 1230	. . . 4 |- ( ph -> ( ph \/ ch \/ th ) )
2	1	imim1i 63	. . 3 |- ( ( ( ph \/ ch \/ th ) -> ps ) -> ( ph -> ps ) )
3		3mix2 1231	. . . 4 |- ( ch -> ( ph \/ ch \/ th ) )
4	3	imim1i 63	. . 3 |- ( ( ( ph \/ ch \/ th ) -> ps ) -> ( ch -> ps ) )
5		3mix3 1232	. . . 4 |- ( th -> ( ph \/ ch \/ th ) )
6	5	imim1i 63	. . 3 |- ( ( ( ph \/ ch \/ th ) -> ps ) -> ( th -> ps ) )
7	2, 4, 6	3jca 1242	. 2 |- ( ( ( ph \/ ch \/ th ) -> ps ) -> ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) )
8		3jao 1389	. 2 |- ( ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) -> ( ( ph \/ ch \/ th ) -> ps ) )
9	7, 8	impbii 199	1 |- ( ( ( ph \/ ch \/ th ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ w3o 1036   /\ w3a 1037

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 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039

This theorem is referenced by: (None)