Metamath Proof Explorer

Theorem adh-minim-ax1

Description: Derivation of ax-1 from adh-minim and ax-mp . Carew Arthur Meredith derived ax-1 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: CpCqp . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion adh-minim-ax1 
|- ( ph -> ( ps -> ph ) )

Proof

Step Hyp Ref Expression
1 adh-minim-ax1-ax2-lem1 
 |-  ( ph -> ( ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) -> ( ps -> ph ) ) )
2 adh-minim-ax1-ax2-lem1 
 |-  ( ( ps -> ph ) -> ( ( ps -> ( ( et -> ( ( ze -> ( ps -> si ) ) -> ( ze -> si ) ) ) -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) -> ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) )
3 adh-minim-ax1-ax2-lem3 
 |-  ( ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ( ps -> ph ) ) -> ( ps -> ( ( et -> ( ( ze -> ( ps -> si ) ) -> ( ze -> si ) ) ) -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) )
4 adh-minim-ax1-ax2-lem4 
 |-  ( ( ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ( ps -> ph ) ) -> ( ps -> ( ( et -> ( ( ze -> ( ps -> si ) ) -> ( ze -> si ) ) ) -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) ) -> ( ( ( ps -> ph ) -> ( ( ps -> ( ( et -> ( ( ze -> ( ps -> si ) ) -> ( ze -> si ) ) ) -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) -> ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) ) -> ( ( ps -> ph ) -> ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ps -> ph ) -> ( ( ps -> ( ( et -> ( ( ze -> ( ps -> si ) ) -> ( ze -> si ) ) ) -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) -> ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) ) -> ( ( ps -> ph ) -> ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) )
6 2 5 ax-mp 
 |-  ( ( ps -> ph ) -> ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) )
7 adh-minim-ax1-ax2-lem4 
 |-  ( ( ( ps -> ph ) -> ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) ) -> ( ( ph -> ( ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) -> ( ps -> ph ) ) ) -> ( ph -> ( ps -> ph ) ) ) )
8 6 7 ax-mp 
 |-  ( ( ph -> ( ( ps -> ( ( ch -> ( ( th -> ( ps -> ta ) ) -> ( th -> ta ) ) ) -> ph ) ) -> ( ps -> ph ) ) ) -> ( ph -> ( ps -> ph ) ) )
9 1 8 ax-mp 
 |-  ( ph -> ( ps -> ph ) )