Theorem sylanblrc 697

Description: Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)

Hypotheses

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

Assertion

Ref	Expression
sylanblrc	|- ( ph -> th )

Proof of Theorem sylanblrc

Step	Hyp	Ref	Expression
1		sylanblrc.1	. 2 |- ( ph -> ps )
2		sylanblrc.2	. 2 |- ch
3		sylanblrc.3	. . 3 |- ( th <-> ( ps /\ ch ) )
4	3	biimpri 218	. 2 |- ( ( ps /\ ch ) -> th )
5	1, 2, 4	sylancl 694	1 |- ( ph -> th )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384

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

This theorem is referenced by:  fnwelem  7292  tfrlem10  7483  gruina  9640  dfac14  21421  1trld  27002  1stmbfm  30322  2ndmbfm  30323  bj-projval  32984  rfcnpre1  39178