Theorem jaao 330

Description: Inference conjoining and disjoining the antecedents of two implications.

Hypotheses

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

Assertion

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

Proof of Theorem jaao

Step	Hyp	Ref	Expression
1		jaao.1	. . 3 |- (ph -> (ps -> ch))
2	1	adantr 306	. 2 |- ((ph /\ th) -> (ps -> ch))
3		jaao.2	. . 3 |- (th -> (ta -> ch))
4	3	adantl 305	. 2 |- ((ph /\ th) -> (ta -> ch))
5	2, 4	jaod 329	1 |- ((ph /\ th) -> ((ps \/ ta ) -> ch))

Syntax hints:   -> wi 2   \/ wo 195   /\ wa 196

This theorem is referenced by:  prss 1854  fr2nr 2177  ordtri1 2231  ordun 2332  suc11 2341  funun 2700  suc11reg 3456  abslt 4855  absle 4856

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

metamath.org