Metamath Proof Explorer

Theorem merlem10

Description: Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merlem10 
|- ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) )

Proof

Step Hyp Ref Expression
1 meredith 
 |-  ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) )
2 meredith 
 |-  ( ( ( ( ( ( ph -> ps ) -> ph ) -> ( -. ph -> -. th ) ) -> ph ) -> ph ) -> ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) )
3 merlem9 
 |-  ( ( ( ( ( ( ( ph -> ps ) -> ph ) -> ( -. ph -> -. th ) ) -> ph ) -> ph ) -> ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) ) -> ( ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) ) )
4 2 3 ax-mp 
 |-  ( ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) )
5 1 4 ax-mp 
 |-  ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) )