Theorem syl2anbr 292

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 285	. 2 |- ( ( ph /\ ch ) -> th )
5	1, 4	sylan2br 288	1 |- ( ( ph /\ ta ) -> th )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  sylancbr  423  tz6.12  5723  ltresr  8206  divmuldivap  9042  fnn0ind  9762  rexanuz  11754  nprmi  12902  cncfval  15673