Theorem syl3an1b 1171

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)

Hypotheses

Ref	Expression
syl3an1b.1	|- ( ph <-> ps )
syl3an1b.2	|- ( ( ps /\ ch /\ th ) -> ta )

Assertion

Ref	Expression
syl3an1b	|- ( ( ph /\ ch /\ th ) -> ta )

Proof of Theorem syl3an1b

Step	Hyp	Ref	Expression
1		syl3an1b.1	. . 3 |- ( ph <-> ps )
2	1	biimpi 113	. 2 |- ( ph -> ps )
3		syl3an1b.2	. 2 |- ( ( ps /\ ch /\ th ) -> ta )
4	2, 3	syl3an1 1168	1 |- ( ( ph /\ ch /\ th ) -> ta )

Syntax hints:   -> wi 4   <-> wb 98   /\ w3a 885

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  df-3an 887

This theorem is referenced by:  irrmul  8581  xrlttr  8716