Theorem syl2anbr 351

Description: A double syllogism inference.

Hypotheses

Ref	Expression
sylan.1	|- ((ph /\ ps) -> ch)
syl2anbr.2	|- (ph <-> th)
syl2anbr.3	|- (ps <-> ta )

Assertion

Ref	Expression
syl2anbr	|- ((th /\ ta ) -> ch)

Proof of Theorem syl2anbr

Step	Hyp	Ref	Expression
1		sylan.1	. . 3 |- ((ph /\ ps) -> ch)
2		syl2anbr.2	. . 3 |- (ph <-> th)
3	1, 2	sylanbr 345	. 2 |- ((th /\ ps) -> ch)
4		syl2anbr.3	. 2 |- (ps <-> ta )
5	3, 4	sylan2br 348	1 |- ((th /\ ta ) -> ch)

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

This theorem is referenced by:  sylancbr 363  cdaen 3719  ltresr 4052  sshjvalt 5321  sshjval3t 5327  hosmvalt 5487  hodmvalt 5488

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