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Metamath Proof Explorer

Theorem merco1lem3

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 17-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merco1lem3 
|- ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) )

Proof

Step Hyp Ref Expression
1 merco1lem2 
 |-  ( ( ( ph -> ph ) -> F. ) -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> F. ) )
2 retbwax2 
 |-  ( ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) -> ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) )
3 merco1lem2 
 |-  ( ( ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) -> ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) ) -> ( ( ( ( ph -> ph ) -> F. ) -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> F. ) ) -> ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) ) )
4 2 3 ax-mp 
 |-  ( ( ( ( ph -> ph ) -> F. ) -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> F. ) ) -> ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) )
5 1 4 ax-mp 
 |-  ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) )
6 merco1lem2 
 |-  ( ( ( ch -> ph ) -> F. ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> F. ) )
7 retbwax2 
 |-  ( ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) -> ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) ) )
8 merco1lem2 
 |-  ( ( ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) -> ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) ) ) -> ( ( ( ( ch -> ph ) -> F. ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> F. ) ) -> ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) ) ) )
9 7 8 ax-mp 
 |-  ( ( ( ( ch -> ph ) -> F. ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> F. ) ) -> ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) ) )
10 6 9 ax-mp 
 |-  ( ( ph -> ( ( ( ph -> ph ) -> ( ph -> F. ) ) -> ( ph -> ph ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) )
11 5 10 ax-mp 
 |-  ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) )