Theorem sylanbr 279

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 131	. 2 |- ( ph -> ps )
3		sylanbr.2	. 2 |- ( ( ps /\ ch ) -> th )
4	2, 3	sylan 277	1 |- ( ( ph /\ ch ) -> th )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  syl2anbr  286  mosubt  2769  xpiindim  4491  funfvdm  5257  caovimo  5714  tfrlem7  5956  iinerm  6201  expclzaplem  9500  expgt0  9509  expge0  9512  expge1  9513  rplpwr  10416