Theorem sylanl1 402

Description: A syllogism inference. (Contributed by NM, 10-Mar-2005.)

Hypotheses

Ref	Expression
sylanl1.1	|- ( ph -> ps )
sylanl1.2	|- ( ( ( ps /\ ch ) /\ th ) -> ta )

Assertion

Ref	Expression
sylanl1	|- ( ( ( ph /\ ch ) /\ th ) -> ta )

Proof of Theorem sylanl1

Step	Hyp	Ref	Expression
1		sylanl1.1	. . 3 |- ( ph -> ps )
2	1	anim1i 340	. 2 |- ( ( ph /\ ch ) -> ( ps /\ ch ) )
3		sylanl1.2	. 2 |- ( ( ( ps /\ ch ) /\ th ) -> ta )
4	2, 3	sylan 283	1 |- ( ( ( ph /\ ch ) /\ 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:  adantlll  480  adantllr  481  adantl3r  512  isocnv  5986  mapxpen  7103  nqnq0pi  7755  nqpnq0nq  7770  addnqprl  7846  addnqpru  7847  pcqmul  13005  infpnlem1  13061  setsn0fun  13266  gsumfzz  13725  dvmptfsum  15607  usgr2edg  16220  usgr2edg1  16222