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Metamath Proof Explorer

Theorem merco1lem2

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 merco1lem2 
|- ( ( ( ph -> ps ) -> ch ) -> ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> ch ) )

Proof

Step Hyp Ref Expression
1 retbwax2 
 |-  ( ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> F. ) -> ( ( ch -> ph ) -> ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> F. ) ) )
2 merco1 
 |-  ( ( ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> F. ) -> ( ( ch -> ph ) -> ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> F. ) ) ) -> ( ( ( ( ch -> ph ) -> ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> F. ) ) -> ps ) -> ( ph -> ps ) ) )
3 1 2 ax-mp 
 |-  ( ( ( ( ch -> ph ) -> ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> F. ) ) -> ps ) -> ( ph -> ps ) )
4 merco1 
 |-  ( ( ( ( ( ch -> ph ) -> ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> F. ) ) -> ps ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> ch ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ph -> ps ) -> ch ) -> ( ( ( ps -> ta ) -> ( ph -> F. ) ) -> ch ) )