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Metamath Proof Explorer

Theorem merlem1

Description: Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merlem1 
|- ( ( ( ch -> ( -. ph -> ps ) ) -> ta ) -> ( ph -> ta ) )

Proof

Step Hyp Ref Expression
1 meredith 
 |-  ( ( ( ( ( -. ph -> ps ) -> ( -. ( -. ta -> -. ch ) -> -. -. ( -. ph -> ps ) ) ) -> ( -. ta -> -. ch ) ) -> ta ) -> ( ( ta -> -. ph ) -> ( -. ( -. ph -> ps ) -> -. ph ) ) )
2 meredith 
 |-  ( ( ( ( ( ( -. ph -> ps ) -> ( -. ( -. ta -> -. ch ) -> -. -. ( -. ph -> ps ) ) ) -> ( -. ta -> -. ch ) ) -> ta ) -> ( ( ta -> -. ph ) -> ( -. ( -. ph -> ps ) -> -. ph ) ) ) -> ( ( ( ( ta -> -. ph ) -> ( -. ( -. ph -> ps ) -> -. ph ) ) -> ( -. ph -> ps ) ) -> ( ch -> ( -. ph -> ps ) ) ) )
3 1 2 ax-mp 
 |-  ( ( ( ( ta -> -. ph ) -> ( -. ( -. ph -> ps ) -> -. ph ) ) -> ( -. ph -> ps ) ) -> ( ch -> ( -. ph -> ps ) ) )
4 meredith 
 |-  ( ( ( ( ( ta -> -. ph ) -> ( -. ( -. ph -> ps ) -> -. ph ) ) -> ( -. ph -> ps ) ) -> ( ch -> ( -. ph -> ps ) ) ) -> ( ( ( ch -> ( -. ph -> ps ) ) -> ta ) -> ( ph -> ta ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ch -> ( -. ph -> ps ) ) -> ta ) -> ( ph -> ta ) )