Theorem jaao 671

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 270	. 2 |- ( ( ph /\ th ) -> ( ps -> ch ) )
3		jaao.2	. . 3 |- ( th -> ( ta -> ch ) )
4	3	adantl 271	. 2 |- ( ( ph /\ th ) -> ( ta -> ch ) )
5	2, 4	jaod 669	1 |- ( ( ph /\ th ) -> ( ( ps \/ ta ) -> ch ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm3.48  731  prlem1  914  nford  1499  funun  4964  poxp  5873  nntri3or  6095