Theorem adh-minimp-sylsimp

Description: Derivation of jarr (also called "syll-simp") from minimp and ax-mp . Polish prefix notation: CCCpqrCqr . (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-sylsimp	|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minimp-jarr-ax2c-lem3	|- ( ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ph -> ps ) -> ch ) )
2		adh-minimp-jarr-imim1-ax2c-lem1	|- ( ( ( ph -> ps ) -> ( ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ph -> ps ) -> ch ) ) ) -> ( ( ( ( ( ph -> ps ) -> ch ) -> ( ph -> ps ) ) -> ( ( ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) ) )
3		adh-minimp-jarr-lem2	|- ( ( ( ( ph -> ps ) -> ( ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ph -> ps ) -> ch ) ) ) -> ( ( ( ( ( ph -> ps ) -> ch ) -> ( ph -> ps ) ) -> ( ( ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) ) ) -> ( ( ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) ) )
4	2 3	ax-mp	|- ( ( ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) -> ( ( ph -> ps ) -> ( ph -> ps ) ) ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ph -> ps ) -> ch ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) )
5	1 4	ax-mp	|- ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) )
6		adh-minimp-jarr-imim1-ax2c-lem1	|- ( ( ( ( ph -> ps ) -> ch ) -> ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) ) -> ( ( ( ps -> ( ( ph -> ps ) -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) -> ( ps -> ch ) ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) ) )
7		adh-minimp-jarr-lem2	|- ( ( ( ( ( ph -> ps ) -> ch ) -> ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) ) -> ( ( ( ps -> ( ( ph -> ps ) -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) -> ( ps -> ch ) ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) ) ) -> ( ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) ) )
8	6 7	ax-mp	|- ( ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ch ) -> ch ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) )
9	5 8	ax-mp	|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )