Theorem sylanr2 385

Description: A syllogism inference. (Contributed by NM, 9-Apr-2005.)

Hypotheses

Ref	Expression
sylanr2.1	|- ( ph -> th )
sylanr2.2	|- ( ( ps /\ ( ch /\ th ) ) -> ta )

Assertion

Ref	Expression
sylanr2	|- ( ( ps /\ ( ch /\ ph ) ) -> ta )

Proof of Theorem sylanr2

Step	Hyp	Ref	Expression
1		sylanr2.1	. . 3 |- ( ph -> th )
2	1	anim2i 324	. 2 |- ( ( ch /\ ph ) -> ( ch /\ th ) )
3		sylanr2.2	. 2 |- ( ( ps /\ ( ch /\ th ) ) -> ta )
4	2, 3	sylan2 270	1 |- ( ( ps /\ ( ch /\ ph ) ) -> ta )

Syntax hints:   -> wi 4   /\ wa 97

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem is referenced by:  adantrrl  455  adantrrr  456  1stconst  5842  2ndconst  5843  ltexprlemopl  6699  ltexprlemopu  6701  mulsub  7398  fzsubel  8923  expsubap  9302