Metamath Proof Explorer

Theorem merlem3

Description: Step 7 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 merlem3 
|- ( ( ( ps -> ch ) -> ph ) -> ( ch -> ph ) )

Proof

Step Hyp Ref Expression
1 merlem2 
 |-  ( ( ( -. ch -> -. ch ) -> ( -. ch -> -. ch ) ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) )
2 merlem2 
 |-  ( ( ( ( -. ch -> -. ch ) -> ( -. ch -> -. ch ) ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) ) -> ( ( ( ( ch -> ph ) -> ( -. ps -> -. ps ) ) -> ps ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) ) )
3 1 2 ax-mp 
 |-  ( ( ( ( ch -> ph ) -> ( -. ps -> -. ps ) ) -> ps ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) )
4 meredith 
 |-  ( ( ( ( ( ch -> ph ) -> ( -. ps -> -. ps ) ) -> ps ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) ) -> ( ( ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) -> ch ) -> ( ps -> ch ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) -> ch ) -> ( ps -> ch ) )
6 meredith 
 |-  ( ( ( ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) -> ch ) -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ph ) -> ( ch -> ph ) ) )
7 5 6 ax-mp 
 |-  ( ( ( ps -> ch ) -> ph ) -> ( ch -> ph ) )