Theorem sylancb 362

Description: A syllogism inference combined with contraction.

Hypotheses

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

Assertion

Ref	Expression
sylancb	|- (th -> ch)

Proof of Theorem sylancb

Step	Hyp	Ref	Expression
1		sylancb.1	. . 3 |- ((ph /\ ps) -> ch)
2		sylancb.2	. . 3 |- (th <-> ph)
3		sylancb.3	. . 3 |- (th <-> ps)
4	1, 2, 3	syl2anb 350	. 2 |- ((th /\ th) -> ch)
5	4	anidms 332	1 |- (th -> ch)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org