Metamath Proof Explorer

Theorem mercolem7

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 . (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion mercolem7 
|- ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) )

Proof

Step Hyp Ref Expression
1 merco2 
 |-  ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) )
2 mercolem3 
 |-  ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) )
3 mercolem6 
 |-  ( ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) ) -> ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) )
4 2 3 ax-mp 
 |-  ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) )
5 mercolem5 
 |-  ( ph -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) )
6 mercolem4 
 |-  ( ( ph -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) ) -> ( ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) ) ) )
7 5 6 ax-mp 
 |-  ( ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) ) )
8 4 7 ax-mp 
 |-  ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) )
9 1 8 ax-mp 
 |-  ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) )