Theorem sylanr2 409

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 342	. 2 |- ( ( ch /\ ph ) -> ( ch /\ th ) )
3		sylanr2.2	. 2 |- ( ( ps /\ ( ch /\ th ) ) -> ta )
4	2, 3	sylan2 286	1 |- ( ( ps /\ ( ch /\ ph ) ) -> ta )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is used by:  adantrrl  490  adantrrr  491  1stconst  6457  2ndconst  6458  ltexprlemopl  7968  ltexprlemopu  7970  mulsub  8728  fzsubel  10466  expsubap  11024  tgcl  15165