Theorem sylanbr 285

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 133	. 2 |- ( ph -> ps )
3		sylanbr.2	. 2 |- ( ( ps /\ ch ) -> th )
4	2, 3	sylan 283	1 |- ( ( ph /\ ch ) -> th )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105

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 depends on definitions:  df-bi 117

This theorem is referenced by:  syl2anbr  292  mosubt  2938  r19.2m  3534  funfvdm  5621  caovimo  6114  tfrlem7  6372  iinerm  6663  expclzaplem  10637  expgt0  10646  expge0  10649  expge1  10650  rplpwr  12167  4sqlem19  12550