Theorem sylanbr 280

Description: A syllogism inference. (Contributed by NM, 18-May-1994.)

Hypotheses

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

Assertion

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

Proof of Theorem sylanbr

Step	Hyp	Ref	Expression
1		sylanbr.1	. . 3 |- ( ps <-> ph )
2	1	biimpri 132	. 2 |- ( ph -> ps )
3		sylanbr.2	. 2 |- ( ( ps /\ ch ) -> th )
4	2, 3	sylan 278	1 |- ( ( ph /\ ch ) -> th )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  syl2anbr  287  mosubt  2793  xpiindim  4586  funfvdm  5380  caovimo  5852  tfrlem7  6096  iinerm  6378  expclzaplem  10033  expgt0  10042  expge0  10045  expge1  10046  rplpwr  11348