Theorem syl2anbr 286

Description: A double syllogism inference. (Contributed by NM, 29-Jul-1999.)

Hypotheses

Ref	Expression
syl2anbr.1	|- ( ps <-> ph )
syl2anbr.2	|- ( ch <-> ta )
syl2anbr.3	|- ( ( ps /\ ch ) -> th )

Assertion

Ref	Expression
syl2anbr	|- ( ( ph /\ ta ) -> th )

Proof of Theorem syl2anbr

Step	Hyp	Ref	Expression
1		syl2anbr.2	. 2 |- ( ch <-> ta )
2		syl2anbr.1	. . 3 |- ( ps <-> ph )
3		syl2anbr.3	. . 3 |- ( ( ps /\ ch ) -> th )
4	2, 3	sylanbr 279	. 2 |- ( ( ph /\ ch ) -> th )
5	1, 4	sylan2br 282	1 |- ( ( ph /\ ta ) -> 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:  sylancbr  410  tz6.12  5222  ltresr  7007  divmuldivap  7800  fnn0ind  8463  rexanuz  9874  nprmi  10506