Theorem syl3an3b 864

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	|- ((ph /\ ps /\ ch) -> th)
syl3an3b.2	|- (ta <-> ch)

Assertion

Ref	Expression
syl3an3b	|- ((ph /\ ps /\ ta) -> th)

Proof of Theorem syl3an3b

Step	Hyp	Ref	Expression
1		syl3an.1	. 2 |- ((ph /\ ps /\ ch) -> th)
2		syl3an3b.2	. . 3 |- (ta <-> ch)
3	2	biimp 151	. 2 |- (ta -> ch)
4	1, 3	syl3an3 861	1 |- ((ph /\ ps /\ ta) -> th)

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 775

This theorem is referenced by:  nnmsucr 4240  xrlttrt 5553  supxrre 6083  basgent 7640  intcld 7680  cdj3lem3 10365

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777

Copyright terms: Public domain