Theorem sylanl2 403

Description: A syllogism inference. (Contributed by NM, 1-Jan-2005.)

Hypotheses

Ref	Expression
sylanl2.1	|- ( ph -> ch )
sylanl2.2	|- ( ( ( ps /\ ch ) /\ th ) -> ta )

Assertion

Ref	Expression
sylanl2	|- ( ( ( ps /\ ph ) /\ th ) -> ta )

Proof of Theorem sylanl2

Step	Hyp	Ref	Expression
1		sylanl2.1	. . 3 |- ( ph -> ch )
2	1	anim2i 342	. 2 |- ( ( ps /\ ph ) -> ( ps /\ ch ) )
3		sylanl2.2	. 2 |- ( ( ( ps /\ ch ) /\ th ) -> ta )
4	2, 3	sylan 283	1 |- ( ( ( ps /\ ph ) /\ th ) -> ta )

Syntax hints:   -> wi 4   /\ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is referenced by:  mpanlr1  440  adantlrl  482  adantlrr  483  cnegexlem3  8220  mulsub  8444  divsubdivap  8772  modqcyc2  10469  lcmneg  12267