Theorem bi3ant 222

Description: Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)

Hypothesis

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

Assertion

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

Proof of Theorem bi3ant

Step	Hyp	Ref	Expression
1		bi1 116	. . 3 |- ( ( th <-> ta ) -> ( th -> ta ) )
2	1	imim1i 59	. 2 |- ( ( ( th -> ta ) -> ph ) -> ( ( th <-> ta ) -> ph ) )
3		bi2 128	. . 3 |- ( ( th <-> ta ) -> ( ta -> th ) )
4	3	imim1i 59	. 2 |- ( ( ( ta -> th ) -> ps ) -> ( ( th <-> ta ) -> ps ) )
5		bi3ant.1	. . 3 |- ( ph -> ( ps -> ch ) )
6	5	imim3i 60	. 2 |- ( ( ( th <-> ta ) -> ph ) -> ( ( ( th <-> ta ) -> ps ) -> ( ( th <-> ta ) -> ch ) ) )
7	2, 4, 6	syl2im 38	1 |- ( ( ( th -> ta ) -> ph ) -> ( ( ( ta -> th ) -> ps ) -> ( ( th <-> ta ) -> ch ) ) )

Syntax hints:   -> wi 4   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bisym  223