Metamath Proof Explorer

Theorem merco1lem12

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 merco1lem12 
|- ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ps ) )

Proof

Step Hyp Ref Expression
1 merco1lem3 
 |-  ( ( ( ( ph -> ta ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> ( ch -> F. ) ) -> ( ch -> ( ph -> ta ) ) )
2 merco1 
 |-  ( ( ( ( ( ph -> ta ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> ( ch -> F. ) ) -> ( ch -> ( ph -> ta ) ) ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) ) )
3 1 2 ax-mp 
 |-  ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) )
4 merco1lem9 
 |-  ( ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) )
5 3 4 ax-mp 
 |-  ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph )
6 merco1lem11 
 |-  ( ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ph ) -> ( ( ( ( ps -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> F. ) -> ph ) )
7 5 6 ax-mp 
 |-  ( ( ( ( ps -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> F. ) -> ph )
8 merco1 
 |-  ( ( ( ( ( ps -> ph ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> F. ) ) -> F. ) -> ph ) -> ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ps ) ) )
9 7 8 ax-mp 
 |-  ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ps ) )