Theorem sylanr1 684

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

Hypotheses

Ref	Expression
sylanr1.1	|- ( ph -> ch )
sylanr1.2	|- ( ( ps /\ ( ch /\ th ) ) -> ta )

Assertion

Ref	Expression
sylanr1	|- ( ( ps /\ ( ph /\ th ) ) -> ta )

Proof of Theorem sylanr1

Step	Hyp	Ref	Expression
1		sylanr1.1	. . 3 |- ( ph -> ch )
2	1	anim1i 592	. 2 |- ( ( ph /\ th ) -> ( ch /\ th ) )
3		sylanr1.2	. 2 |- ( ( ps /\ ( ch /\ th ) ) -> ta )
4	2, 3	sylan2 491	1 |- ( ( ps /\ ( ph /\ th ) ) -> 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:  adantrll  758  adantrlr  759  sbthlem9  8078  pczpre  15552  cpmadugsumlemF  20681  blsscls2  22309  rpvmasumlem  25176  leopmuli  28992  chirredlem1  29249  chirredlem3  29251  dvconstbi  38533  bccbc  38544  reccot  42499  rectan  42500  aacllem  42547