Theorem jaao 495

Description: Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)

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 451	. 2 |- ((ph /\ th) -> (ps -> ch))
3		jaao.2	. . 3 |- (th -> (ta -> ch))
4	3	adantl 452	. 2 |- ((ph /\ th) -> (ta -> ch))
5	2, 4	jaod 369	1 |- ((ph /\ th) -> ((ps \/ ta) -> ch))

This proof depends on syntax axioms:   -> wi 4   \/ wo 357   /\ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by:  pm3.44  497  pm3.48  806  prlem1  928  funun  5147