Theorem 3jaao 1298

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 1008	. 2 |- ( ( ph /\ th /\ et ) -> ( ps -> ch ) )
3		3jaao.2	. . 3 |- ( th -> ( ta -> ch ) )
4	3	3ad2ant2 1009	. 2 |- ( ( ph /\ th /\ et ) -> ( ta -> ch ) )
5		3jaao.3	. . 3 |- ( et -> ( ze -> ch ) )
6	5	3ad2ant3 1010	. 2 |- ( ( ph /\ th /\ et ) -> ( ze -> ch ) )
7	2, 4, 6	3jaod 1294	1 |- ( ( ph /\ th /\ et ) -> ( ( ps \/ ta \/ ze ) -> ch ) )

Syntax hints:   -> wi 4   \/ w3o 967   /\ w3a 968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970

This theorem is referenced by: (None)