Theorem syl3anr1 1378

Description: A syllogism inference. (Contributed by NM, 31-Jul-2007.)

Hypotheses

Ref	Expression
syl3anr1.1	|- ( ph -> ps )
syl3anr1.2	|- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et )

Assertion

Ref	Expression
syl3anr1	|- ( ( ch /\ ( ph /\ th /\ ta ) ) -> et )

Proof of Theorem syl3anr1

Step	Hyp	Ref	Expression
1		syl3anr1.1	. . 3 |- ( ph -> ps )
2	1	3anim1i 1248	. 2 |- ( ( ph /\ th /\ ta ) -> ( ps /\ th /\ ta ) )
3		syl3anr1.2	. 2 |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et )
4	2, 3	sylan2 491	1 |- ( ( ch /\ ( ph /\ th /\ ta ) ) -> et )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  btwnconn1lem4  32197  pridlc2  33871  atmod1i1  35143  prmdvdsfmtnof1lem2  41497