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Theorem mercolem3 1504
 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
mercolem3 ((ψχ) → (ψ → (φχ)))

Proof of Theorem mercolem3
StepHypRef Expression
1 merco2 1501 . 2 (((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ))))
2 merco2 1501 . . . 4 (((χφ) → (( ⊥ → φ) → ψ)) → ((ψχ) → (ψ → (φχ))))
3 mercolem2 1503 . . . . . . 7 (((ψ → (φχ)) → ψ) → (( ⊥ → φ) → (( ⊥ → φ) → ψ)))
4 merco2 1501 . . . . . . 7 ((((ψ → (φχ)) → ψ) → (( ⊥ → φ) → (( ⊥ → φ) → ψ))) → (((( ⊥ → φ) → ψ) → (ψ → (φχ))) → (( ⊥ → φ) → ((ψχ) → (ψ → (φχ))))))
53, 4ax-mp 8 . . . . . 6 (((( ⊥ → φ) → ψ) → (ψ → (φχ))) → (( ⊥ → φ) → ((ψχ) → (ψ → (φχ)))))
6 merco2 1501 . . . . . 6 ((((( ⊥ → φ) → ψ) → (ψ → (φχ))) → (( ⊥ → φ) → ((ψχ) → (ψ → (φχ))))) → ((((ψχ) → (ψ → (φχ))) → (( ⊥ → φ) → ψ)) → (( ⊥ → φ) → ((χφ) → (( ⊥ → φ) → ψ)))))
75, 6ax-mp 8 . . . . 5 ((((ψχ) → (ψ → (φχ))) → (( ⊥ → φ) → ψ)) → (( ⊥ → φ) → ((χφ) → (( ⊥ → φ) → ψ))))
8 merco2 1501 . . . . 5 (((((ψχ) → (ψ → (φχ))) → (( ⊥ → φ) → ψ)) → (( ⊥ → φ) → ((χφ) → (( ⊥ → φ) → ψ)))) → ((((χφ) → (( ⊥ → φ) → ψ)) → ((ψχ) → (ψ → (φχ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((ψχ) → (ψ → (φχ)))))))
97, 8ax-mp 8 . . . 4 ((((χφ) → (( ⊥ → φ) → ψ)) → ((ψχ) → (ψ → (φχ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((ψχ) → (ψ → (φχ))))))
102, 9ax-mp 8 . . 3 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((ψχ) → (ψ → (φχ)))))
111, 10ax-mp 8 . 2 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((ψχ) → (ψ → (φχ))))
121, 11ax-mp 8 1 ((ψχ) → (ψ → (φχ)))
 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:  mercolem4  1505  mercolem7  1508  mercolem8  1509  re1tbw1  1510  re1tbw4  1513
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