Theorem adh-minimp-ax2

Description: Derivation of ax-2 from adh-minimp and ax-mp . Polish prefix notation: CCpCqrCCpqCpr . (Contributed by BJ, 4-Apr-2021) (Revised by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	adh-minimp-ax2	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minimp-ax2-lem4	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
2		adh-minimp-ax2c	|- ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) )
3		adh-minimp-ax2-lem4	|- ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) )
4	2 3	ax-mp	|- ( ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
5	1 4	ax-mp	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )