Theorem sylanr2 685

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 593	. 2 |- ( ( ch /\ ph ) -> ( ch /\ th ) )
3		sylanr2.2	. 2 |- ( ( ps /\ ( ch /\ th ) ) -> ta )
4	2, 3	sylan2 491	1 |- ( ( ps /\ ( ch /\ ph ) ) -> ta )

Syntax hints:   -> wi 4   /\ wa 384

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 197  df-an 386

This theorem is referenced by:  adantrrl  760  adantrrr  761  1stconst  7265  2ndconst  7266  isfin7-2  9218  mulsub  10473  fzsubel  12377  expsub  12908  ramlb  15723  0ram  15724  ressmplvsca  19459  tgcl  20773  fgss2  21678  nmoid  22546  chirredlem4  29252  poimirlem28  33437  pridlc3  33872  stoweidlem34  40251