Theorem antnestlaw3

Description: A law of nested antecedents. Compare with looinv . (Contributed by Adrian Ducourtial, 5-Dec-2025)

	Ref	Expression
Assertion	antnestlaw3	|- ( ( ( ( ph -> ps ) -> ch ) -> ch ) <-> ( ( ( ph -> ch ) -> ps ) -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		antnestlaw3lem	|- ( -. ( ( ( ph -> ch ) -> ps ) -> ps ) -> -. ( ( ( ph -> ps ) -> ch ) -> ch ) )
2	1	con4i	|- ( ( ( ( ph -> ps ) -> ch ) -> ch ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) )
3		antnestlaw3lem	|- ( -. ( ( ( ph -> ps ) -> ch ) -> ch ) -> -. ( ( ( ph -> ch ) -> ps ) -> ps ) )
4	3	con4i	|- ( ( ( ( ph -> ch ) -> ps ) -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) )
5	2 4	impbii	|- ( ( ( ( ph -> ps ) -> ch ) -> ch ) <-> ( ( ( ph -> ch ) -> ps ) -> ps ) )