Metamath Proof Explorer

Theorem merco1lem10

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 merco1lem10 
|- ( ( ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) -> ph ) -> ( th -> ph ) )

Proof

Step Hyp Ref Expression
1 merco1 
 |-  ( ( ( ( ( ch -> ph ) -> ( ta -> F. ) ) -> ph ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) )
2 merco1lem2 
 |-  ( ( ( ( ( ( ch -> ph ) -> ( ta -> F. ) ) -> ph ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) ) -> ( ( ( ( ph -> ps ) -> ( th -> F. ) ) -> ( ( ( ( ch -> ph ) -> ( ta -> F. ) ) -> ph ) -> F. ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) ) )
3 1 2 ax-mp 
 |-  ( ( ( ( ph -> ps ) -> ( th -> F. ) ) -> ( ( ( ( ch -> ph ) -> ( ta -> F. ) ) -> ph ) -> F. ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) )
4 merco1 
 |-  ( ( ( ( ( ph -> ps ) -> ( th -> F. ) ) -> ( ( ( ( ch -> ph ) -> ( ta -> F. ) ) -> ph ) -> F. ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) ) -> ( ( ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) -> ph ) -> ( th -> ph ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) -> ph ) -> ( th -> ph ) )