Theorem sylanl2 407

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 )

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:  mpanlr1  444  adantlrl  486  adantlrr  487  cnegexlem3  8503  mulsub  8728  divsubdivap  9058  modqcyc2  10797  lcmneg  12852