Theorem bj-bi3ant 31165

Description: This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)

Hypothesis

Ref	Expression
bj-bi3ant.1	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
bj-bi3ant	|- ( ( ( th -> ta ) -> ph ) -> ( ( ( ta -> th ) -> ps ) -> ( ( th <-> ta ) -> ch ) ) )

Proof of Theorem bj-bi3ant

Step	Hyp	Ref	Expression
1		biimp 196	. . 3 |- ( ( th <-> ta ) -> ( th -> ta ) )
2	1	imim1i 60	. 2 |- ( ( ( th -> ta ) -> ph ) -> ( ( th <-> ta ) -> ph ) )
3		biimpr 201	. . 3 |- ( ( th <-> ta ) -> ( ta -> th ) )
4	3	imim1i 60	. 2 |- ( ( ( ta -> th ) -> ps ) -> ( ( th <-> ta ) -> ps ) )
5		bj-bi3ant.1	. . 3 |- ( ph -> ( ps -> ch ) )
6	5	imim3i 61	. 2 |- ( ( ( th <-> ta ) -> ph ) -> ( ( ( th <-> ta ) -> ps ) -> ( ( th <-> ta ) -> ch ) ) )
7	2, 4, 6	syl2im 39	1 |- ( ( ( th -> ta ) -> ph ) -> ( ( ( ta -> th ) -> ps ) -> ( ( th <-> ta ) -> ch ) ) )

Syntax hints:   -> wi 4   <-> wb 187

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 188

This theorem is referenced by:  bj-bisym  31166