Theorem sylancb 646

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 465	. 2 |- ((ph /\ ph) -> th)
5	4	anidms 626	1 |- (ph -> th)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by: (None)