Theorem 3jaao 1249

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 976	. 2 |- ((ph /\ th /\ et) -> (ps -> ch))
3		3jaao.2	. . 3 |- (th -> (ta -> ch))
4	3	3ad2ant2 977	. 2 |- ((ph /\ th /\ et) -> (ta -> ch))
5		3jaao.3	. . 3 |- (et -> (ze -> ch))
6	5	3ad2ant3 978	. 2 |- ((ph /\ th /\ et) -> (ze -> ch))
7	2, 4, 6	3jaod 1246	1 |- ((ph /\ th /\ et) -> ((ps \/ ta \/ ze) -> ch))

Syntax hints:   -> wi 4   \/ 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)