Theorem im2anan9r 563

Description: Deduction joining nested implications to form implication of conjunctions.

Hypotheses

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

Assertion

Ref	Expression
im2anan9r	|- ((th /\ ph) -> ((ps /\ ta) -> (ch /\ et)))

Proof of Theorem im2anan9r

Step	Hyp	Ref	Expression
1		im2an9.1	. . 3 |- (ph -> (ps -> ch))
2	1	adantl 388	. 2 |- ((th /\ ph) -> (ps -> ch))
3		im2an9.2	. . 3 |- (th -> (ta -> et))
4	3	adantr 389	. 2 |- ((th /\ ph) -> (ta -> et))
5	2, 4	anim12d 557	1 |- ((th /\ ph) -> ((ps /\ ta) -> (ch /\ et)))

Syntax hints:   -> wi 3   /\ wa 223

This theorem is referenced by:  pssnn 4519  lbreu 6000  spwmo 8598

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain