Theorem jaao 531

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 481	. 2 |- ( ( ph /\ th ) -> ( ps -> ch ) )
3		jaao.2	. . 3 |- ( th -> ( ta -> ch ) )
4	3	adantl 482	. 2 |- ( ( ph /\ th ) -> ( ta -> ch ) )
5	2, 4	jaod 395	1 |- ( ( ph /\ th ) -> ( ( ps \/ ta ) -> ch ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  pm3.44  533  pm3.48  878  prlem1  1005  ordtri1  5756  ordun  5829  suc11  5831  funun  5932  poxp  7289  suc11reg  8516  rankunb  8713  gruun  9628  ofpreima2  29466  wl-orel12  33294  clsk1indlem3  38341