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Metamath Proof Explorer

Theorem merlem12

Description: Step 28 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 merlem12 
|- ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph )

Proof

Step Hyp Ref Expression
1 merlem5 
 |-  ( ( ch -> ch ) -> ( -. -. ch -> ch ) )
2 merlem2 
 |-  ( ( ( ch -> ch ) -> ( -. -. ch -> ch ) ) -> ( th -> ( -. -. ch -> ch ) ) )
3 1 2 ax-mp 
 |-  ( th -> ( -. -. ch -> ch ) )
4 merlem4 
 |-  ( ( th -> ( -. -. ch -> ch ) ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) ) )
5 3 4 ax-mp 
 |-  ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) )
6 merlem11 
 |-  ( ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) )
7 5 6 ax-mp 
 |-  ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph )