Theorem 3jaoian 1393

Description: Disjunction of three antecedents (inference). (Contributed by NM, 14-Oct-2005.)

Hypotheses

Ref	Expression
3jaoian.1	|- ( ( ph /\ ps ) -> ch )
3jaoian.2	|- ( ( th /\ ps ) -> ch )
3jaoian.3	|- ( ( ta /\ ps ) -> ch )

Assertion

Ref	Expression
3jaoian	|- ( ( ( ph \/ th \/ ta ) /\ ps ) -> ch )

Proof of Theorem 3jaoian

Step	Hyp	Ref	Expression
1		3jaoian.1	. . . 4 |- ( ( ph /\ ps ) -> ch )
2	1	ex 450	. . 3 |- ( ph -> ( ps -> ch ) )
3		3jaoian.2	. . . 4 |- ( ( th /\ ps ) -> ch )
4	3	ex 450	. . 3 |- ( th -> ( ps -> ch ) )
5		3jaoian.3	. . . 4 |- ( ( ta /\ ps ) -> ch )
6	5	ex 450	. . 3 |- ( ta -> ( ps -> ch ) )
7	2, 4, 6	3jaoi 1391	. 2 |- ( ( ph \/ th \/ ta ) -> ( ps -> ch ) )
8	7	imp 445	1 |- ( ( ( ph \/ th \/ ta ) /\ ps ) -> ch )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036

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:  xrltnsym  11970  xrlttri  11972  xrlttr  11973  qbtwnxr  12031  xltnegi  12047  xaddcom  12071  xnegdi  12078  lcmftp  15349  xaddeq0  29518  3ccased  31600