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Metamath Proof Explorer

Theorem mercolem5

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 mercolem5 
|- ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) )

Proof

Step Hyp Ref Expression
1 merco2 
 |-  ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) )
2 merco2 
 |-  ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> th ) ) -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) )
3 mercolem1 
 |-  ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> th ) ) -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) -> ( ( ( F. -> ph ) -> th ) -> ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) ) )
4 2 3 ax-mp 
 |-  ( ( ( F. -> ph ) -> th ) -> ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) )
5 mercolem2 
 |-  ( ( ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) -> th ) -> ( ( F. -> ph ) -> ( ( F. -> ph ) -> th ) ) )
6 merco2 
 |-  ( ( ( ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) -> th ) -> ( ( F. -> ph ) -> ( ( F. -> ph ) -> th ) ) ) -> ( ( ( ( F. -> ph ) -> th ) -> ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) ) ) ) )
7 5 6 ax-mp 
 |-  ( ( ( ( F. -> ph ) -> th ) -> ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) ) ) )
8 4 7 ax-mp 
 |-  ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) ) )
9 1 8 ax-mp 
 |-  ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) )
10 1 9 ax-mp 
 |-  ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) )