Theorem im2anan9 808

Description: Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)

Hypotheses

Ref	Expression
im2an9.1	|- (ph -> (ps -> ch))
im2an9.2	|- (th -> (ta -> et))

Assertion

Ref	Expression
im2anan9	|- ((ph /\ th) -> ((ps /\ ta) -> (ch /\ et)))

Proof of Theorem im2anan9

Step	Hyp	Ref	Expression
1		im2an9.1	. . 3 |- (ph -> (ps -> ch))
2	1	adantr 451	. 2 |- ((ph /\ th) -> (ps -> ch))
3		im2an9.2	. . 3 |- (th -> (ta -> et))
4	3	adantl 452	. 2 |- ((ph /\ th) -> (ta -> et))
5	2, 4	anim12d 546	1 |- ((ph /\ th) -> ((ps /\ ta) -> (ch /\ et)))

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:  im2anan9r  809  ax11eq  2193  ax11el  2194  xpss12  4856  f1oun  5305  fntxp  5805  fnpprod  5844  fce  6189