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Theorem mercolem7 1508
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mercolem7 ((φψ) → (((φχ) → (θψ)) → (θψ)))

Proof of Theorem mercolem7
StepHypRef Expression
1 merco2 1501 . 2 (((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ))))
2 mercolem3 1504 . . . 4 (((φχ) → (θψ)) → ((φχ) → (((φχ) → (θψ)) → (θψ))))
3 mercolem6 1507 . . . 4 ((((φχ) → (θψ)) → ((φχ) → (((φχ) → (θψ)) → (θψ)))) → ((φχ) → (((φχ) → (θψ)) → (θψ))))
42, 3ax-mp 8 . . 3 ((φχ) → (((φχ) → (θψ)) → (θψ)))
5 mercolem5 1506 . . . 4 (φ → ((φψ) → (((φχ) → (θψ)) → (θψ))))
6 mercolem4 1505 . . . 4 ((φ → ((φψ) → (((φχ) → (θψ)) → (θψ)))) → (((φχ) → (((φχ) → (θψ)) → (θψ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φψ) → (((φχ) → (θψ)) → (θψ))))))
75, 6ax-mp 8 . . 3 (((φχ) → (((φχ) → (θψ)) → (θψ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φψ) → (((φχ) → (θψ)) → (θψ)))))
84, 7ax-mp 8 . 2 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φψ) → (((φχ) → (θψ)) → (θψ))))
91, 8ax-mp 8 1 ((φψ) → (((φχ) → (θψ)) → (θψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊥ wfal 1317 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320 This theorem is referenced by:  mercolem8  1509
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