Theorem adh-minimp-jarr-imim1-ax2c-lem1

Description: First lemma for the derivation of jarr , imim1 , and a commuted form of ax-2 , and indirectly ax-1 and ax-2 , from adh-minimp and ax-mp . Polish prefix notation: CCpqCCCrpCqsCps . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	adh-minimp-jarr-imim1-ax2c-lem1	|- ( ( ph -> ps ) -> ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minimp	|- ( et -> ( ( ze -> si ) -> ( ( ( rh -> ze ) -> ( si -> mu ) ) -> ( ze -> mu ) ) ) )
2		adh-minimp	|- ( ( et -> ( ( ze -> si ) -> ( ( ( rh -> ze ) -> ( si -> mu ) ) -> ( ze -> mu ) ) ) ) -> ( ( ph -> ps ) -> ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) )
3	1 2	ax-mp	|- ( ( ph -> ps ) -> ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) )