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Metamath Proof Explorer

Theorem merco1lem14

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 18-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merco1lem14 
|- ( ( ( ( ph -> ps ) -> ps ) -> ch ) -> ( ph -> ch ) )

Proof

Step Hyp Ref Expression
1 merco1lem13 
 |-  ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) )
2 merco1lem8 
 |-  ( ( ( ( ( ph -> ( ( ph -> ps ) -> ps ) ) -> ph ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> F. ) ) -> ph ) -> ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) )
3 merco1 
 |-  ( ( ( ( ( ( ph -> ( ( ph -> ps ) -> ps ) ) -> ph ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> F. ) ) -> ph ) -> ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) ) )
4 2 3 ax-mp 
 |-  ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) )
5 merco1lem9 
 |-  ( ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) )
6 4 5 ax-mp 
 |-  ( ( ( ( ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ps ) ) )
7 1 6 ax-mp 
 |-  ( ph -> ( ( ph -> ps ) -> ps ) )
8 merco1lem12 
 |-  ( ( ph -> ( ( ph -> ps ) -> ps ) ) -> ( ( ( ( ch -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ps ) ) )
9 7 8 ax-mp 
 |-  ( ( ( ( ch -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ps ) )
10 merco1 
 |-  ( ( ( ( ( ch -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ch ) -> ( ph -> ch ) ) )
11 9 10 ax-mp 
 |-  ( ( ( ( ph -> ps ) -> ps ) -> ch ) -> ( ph -> ch ) )