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Metamath Proof Explorer

Theorem mercolem1

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 mercolem1 
|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) )

Proof

Step Hyp Ref Expression
1 merco2 
 |-  ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) )
2 merco2 
 |-  ( ( ( ch -> ph ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) )
3 merco2 
 |-  ( ( ( ps -> ( th -> ch ) ) -> ( ( F. -> ph ) -> F. ) ) -> ( ( F. -> ps ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) )
4 merco2 
 |-  ( ( ( ( ps -> ( th -> ch ) ) -> ( ( F. -> ph ) -> F. ) ) -> ( ( F. -> ps ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) ) -> ( ( ( ( F. -> ph ) -> ( ph -> ps ) ) -> ( ps -> ( th -> ch ) ) ) -> ( ( F. -> ph ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ( F. -> ph ) -> ( ph -> ps ) ) -> ( ps -> ( th -> ch ) ) ) -> ( ( F. -> ph ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) ) )
6 merco2 
 |-  ( ( ( ( ( F. -> ph ) -> ( ph -> ps ) ) -> ( ps -> ( th -> ch ) ) ) -> ( ( F. -> ph ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) ) ) -> ( ( ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) -> ( ( F. -> ph ) -> ( ( ch -> ph ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) ) ) )
7 5 6 ax-mp 
 |-  ( ( ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) -> ( ( F. -> ph ) -> ( ( ch -> ph ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) ) )
8 merco2 
 |-  ( ( ( ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) -> ( ( F. -> ph ) -> ( ( ch -> ph ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) ) ) -> ( ( ( ( ch -> ph ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) ) ) ) )
9 7 8 ax-mp 
 |-  ( ( ( ( ch -> ph ) -> ( ( F. -> ph ) -> ( ph -> ps ) ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) ) ) )
10 2 9 ax-mp 
 |-  ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) ) )
11 1 10 ax-mp 
 |-  ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) )
12 1 11 ax-mp 
 |-  ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) )