Theorem im2anan9 562

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 270	. 2 |- ( ( ph /\ th ) -> ( ps -> ch ) )
3		im2an9.2	. . 3 |- ( th -> ( ta -> et ) )
4	3	adantl 271	. 2 |- ( ( ph /\ th ) -> ( ta -> et ) )
5	2, 4	anim12d 328	1 |- ( ( ph /\ th ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) )

Syntax hints:   -> wi 4   /\ wa 102

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

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  im2anan9r  563  trin  3885  xpss12  4463  f1oun  5166  poxp  5873  brecop  6219  enq0sym  6622  genpdisj  6713