Theorem sylanl1 406

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 )

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:  adantlll  484  adantllr  485  adantl3r  516  isocnv  6017  mapxpen  7148  nqnq0pi  7805  nqpnq0nq  7820  addnqprl  7896  addnqpru  7897  pcqmul  13082  infpnlem1  13138  setsn0fun  13389  dvmptfsum  15826  usgr2edg  16449  usgr2edg1  16451