Theorem sylancb 395

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

Hypotheses

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

Assertion

Ref	Expression
sylancb	|- ( ph -> th )

Proof of Theorem sylancb

Step	Hyp	Ref	Expression
1		sylancb.1	. . 3 |- ( ph <-> ps )
2		sylancb.2	. . 3 |- ( ph <-> ch )
3		sylancb.3	. . 3 |- ( ( ps /\ ch ) -> th )
4	1, 2, 3	syl2anb 275	. 2 |- ( ( ph /\ ph ) -> th )
5	4	anidms 377	1 |- ( ph -> th )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)