Theorem syldan 456

Description: A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)

Hypotheses

Ref	Expression
syldan.1	|- ((ph /\ ps) -> ch)
syldan.2	|- ((ph /\ ch) -> th)

Assertion

Ref	Expression
syldan	|- ((ph /\ ps) -> th)

Proof of Theorem syldan

Step	Hyp	Ref	Expression
1		syldan.1	. 2 |- ((ph /\ ps) -> ch)
2		syldan.2	. . . 4 |- ((ph /\ ch) -> th)
3	2	expcom 424	. . 3 |- (ch -> (ph -> th))
4	3	adantrd 454	. 2 |- (ch -> ((ph /\ ps) -> th))
5	1, 4	mpcom 32	1 |- ((ph /\ ps) -> th)

Syntax hints:   -> wi 4   /\ wa 358

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 177  df-an 360

This theorem is referenced by:  sylan2  460  sbcied2  3083  csbied2  3179  lefinaddc  4450  vfinncvntsp  4549  addlec  6208  nchoicelem5  6293