Theorem minimp-ax2

Description: Derivation of ax-2 from ax-mp and minimp . (Contributed by BJ, 4-Apr-2021) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		minimp-ax2c	|- ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) )
2		minimp-ax2c	|- ( ( ( ph -> ps ) -> ( ph -> ( ps -> ch ) ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
3		minimp-syllsimp	|- ( ( ( ( ph -> ps ) -> ( ph -> ( ps -> ch ) ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) )
4	2 3	ax-mp	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
5		minimp-ax2c	|- ( ( ( ph -> ( ps -> ch ) ) -> ( ( 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 ) ) ) ) )
6		minimp-syllsimp	|- ( ( ( ( ph -> ( ps -> ch ) ) -> ( ( 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 ) ) ) ) ) -> ( ( ( 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 ) ) ) ) ) )
7	5 6	ax-mp	|- ( ( ( 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 ) ) ) ) )
8	1 4 7	mp2	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )