Theorem sylancbr 419

Description: A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)

Hypotheses

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

Assertion

Ref	Expression
sylancbr	|- ( ph -> th )

Proof of Theorem sylancbr

Step	Hyp	Ref	Expression
1		sylancbr.1	. . 3 |- ( ps <-> ph )
2		sylancbr.2	. . 3 |- ( ch <-> ph )
3		sylancbr.3	. . 3 |- ( ( ps /\ ch ) -> th )
4	1, 2, 3	syl2anbr 292	. 2 |- ( ( ph /\ ph ) -> th )
5	4	anidms 397	1 |- ( ph -> 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: (None)