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Metamath Proof Explorer

Theorem merco1lem7

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 merco1lem7 
|- ( ph -> ( ( ( ps -> ch ) -> ps ) -> ps ) )

Proof

Step Hyp Ref Expression
1 merco1lem5 
 |-  ( ( ( ( ps -> F. ) -> ( ( ( ps -> ch ) -> ps ) -> F. ) ) -> ch ) -> ( ps -> ch ) )
2 merco1 
 |-  ( ( ( ( ( ps -> F. ) -> ( ( ( ps -> ch ) -> ps ) -> F. ) ) -> ch ) -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ps ) -> ( ( ( ps -> ch ) -> ps ) -> ps ) ) )
3 1 2 ax-mp 
 |-  ( ( ( ps -> ch ) -> ps ) -> ( ( ( ps -> ch ) -> ps ) -> ps ) )
4 merco1lem6 
 |-  ( ( ( ( ps -> ch ) -> ps ) -> ( ( ( ps -> ch ) -> ps ) -> ps ) ) -> ( ph -> ( ( ( ps -> ch ) -> ps ) -> ps ) ) )
5 3 4 ax-mp 
 |-  ( ph -> ( ( ( ps -> ch ) -> ps ) -> ps ) )