Theorem adh-minim-ax2

Description: Derivation of ax-2 from adh-minim and ax-mp . Carew Arthur Meredith derived ax-2 inA single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas,On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax2c	|- ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) )
2		adh-minim-ax1-ax2-lem3	|- ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ( ( th -> ta ) -> et ) -> ( ( ta -> ( et -> ze ) ) -> ( ta -> ze ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) )
3	1 2	ax-mp	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ( ( th -> ta ) -> et ) -> ( ( ta -> ( et -> ze ) ) -> ( ta -> ze ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
4		adh-minim-ax2-lem6	|- ( ( ( ph -> ( ps -> ch ) ) -> ( ( ( ( th -> ta ) -> et ) -> ( ( ta -> ( et -> ze ) ) -> ( ta -> ze ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
5	3 4	ax-mp	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )