Theorem antnestlaw1

Description: A law of nested antecedents. The converse direction is a subschema of pm2.27 . (Contributed by Adrian Ducourtial, 5-Dec-2025)

	Ref	Expression
Assertion	antnestlaw1	|- ( ( ( ( ph -> ps ) -> ps ) -> ps ) <-> ( ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		pm2.21	|- ( -. ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) )
2		conax1	|- ( -. ( ph -> ps ) -> -. ps )
3	1 2	jcnd	|- ( -. ( ph -> ps ) -> -. ( ( ( ph -> ps ) -> ps ) -> ps ) )
4	3	con4i	|- ( ( ( ( ph -> ps ) -> ps ) -> ps ) -> ( ph -> ps ) )
5		pm2.27	|- ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ps ) -> ps ) )
6	4 5	impbii	|- ( ( ( ( ph -> ps ) -> ps ) -> ps ) <-> ( ph -> ps ) )