Theorem jaao 709

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 274	. 2 |- ( ( ph /\ th ) -> ( ps -> ch ) )
3		jaao.2	. . 3 |- ( th -> ( ta -> ch ) )
4	3	adantl 275	. 2 |- ( ( ph /\ th ) -> ( ta -> ch ) )
5	2, 4	jaod 707	1 |- ( ( ph /\ th ) -> ( ( ps \/ ta ) -> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698

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

This theorem is referenced by:  pm3.48  775  prlem1  963  nford  1555  funun  5232  poxp  6200  nntri3or  6461