Theorem sylanblc 412

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

Hypotheses

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

Assertion

Ref	Expression
sylanblc	|- ( ph -> th )

Proof of Theorem sylanblc

Step	Hyp	Ref	Expression
1		sylanblc.1	. 2 |- ( ph -> ps )
2		sylanblc.2	. 2 |- ch
3		sylanblc.3	. . 3 |- ( ( ps /\ ch ) <-> th )
4	3	biimpi 119	. 2 |- ( ( ps /\ ch ) -> th )
5	1, 2, 4	sylancl 410	1 |- ( ph -> th )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)