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Theorem mercolem7 1668
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1661. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mercolem7 |- ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) )
Proof of Theorem mercolem7
StepHypRef Expression
1 merco2 1661 . 2 |- ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) )
2 mercolem3 1664 . . . 4 |- ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) )
3 mercolem6 1667 . . . 4 |- ( ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) ) -> ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) )
42, 3ax-mp 5 . . 3 |- ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) )
5 mercolem5 1666 . . . 4 |- ( ph -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) )
6 mercolem4 1665 . . . 4 |- ( ( ph -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) ) -> ( ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) ) ) )
75, 6ax-mp 5 . . 3 |- ( ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) -> ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) ) )
84, 7ax-mp 5 . 2 |- ( ( ( ( ph -> ph ) -> ( ( F. -> ph ) -> ph ) ) -> ( ( ph -> ph ) -> ( ph -> ( ph -> ph ) ) ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) ) )
91, 8ax-mp 5 1 |- ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   F. wfal 1488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-tru 1486  df-fal 1489
This theorem is referenced by:  mercolem8  1669
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