Metamath Proof Explorer

Theorem merco1lem13

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 merco1lem13 
|- ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ta ) -> ( ph -> ta ) )

Proof

Step Hyp Ref Expression
1 merco1 
 |-  ( ( ( ( ( ps -> ph ) -> ( ch -> F. ) ) -> ph ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) )
2 merco1lem4 
 |-  ( ( ( ( ( ( ps -> ph ) -> ( ch -> F. ) ) -> ph ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) -> ( ph -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) )
3 1 2 ax-mp 
 |-  ( ph -> ( ( ph -> ps ) -> ( ch -> ps ) ) )
4 merco1lem12 
 |-  ( ( ph -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) -> ( ( ( ( ta -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ( ta -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) )
6 merco1 
 |-  ( ( ( ( ( ta -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) -> ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ta ) -> ( ph -> ta ) ) )
7 5 6 ax-mp 
 |-  ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ta ) -> ( ph -> ta ) )