Theorem 3jaao 1396

Description: Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Hypotheses

Ref	Expression
3jaao.1	|- ( ph -> ( ps -> ch ) )
3jaao.2	|- ( th -> ( ta -> ch ) )
3jaao.3	|- ( et -> ( ze -> ch ) )

Assertion

Ref	Expression
3jaao	|- ( ( ph /\ th /\ et ) -> ( ( ps \/ ta \/ ze ) -> ch ) )

Proof of Theorem 3jaao

Step	Hyp	Ref	Expression
1		3jaao.1	. . 3 |- ( ph -> ( ps -> ch ) )
2	1	3ad2ant1 1082	. 2 |- ( ( ph /\ th /\ et ) -> ( ps -> ch ) )
3		3jaao.2	. . 3 |- ( th -> ( ta -> ch ) )
4	3	3ad2ant2 1083	. 2 |- ( ( ph /\ th /\ et ) -> ( ta -> ch ) )
5		3jaao.3	. . 3 |- ( et -> ( ze -> ch ) )
6	5	3ad2ant3 1084	. 2 |- ( ( ph /\ th /\ et ) -> ( ze -> ch ) )
7	2, 4, 6	3jaod 1392	1 |- ( ( ph /\ th /\ et ) -> ( ( ps \/ ta \/ ze ) -> ch ) )

Syntax hints:   -> wi 4   \/ 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:  lpni  27332  3ornot23  38715