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Theorem merlem7 1413
 Description: Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem7 (φ → (((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ)))

Proof of Theorem merlem7
StepHypRef Expression
1 merlem4 1410 . 2 ((ψχ) → (((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ)))
2 merlem6 1412 . . . 4 ((((χτ) → (¬ θ → ¬ ψ)) → θ) → (((((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ)) → ¬ φ) → (¬ χ → ¬ φ)))
3 ax-meredith 1406 . . . 4 (((((χτ) → (¬ θ → ¬ ψ)) → θ) → (((((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ)) → ¬ φ) → (¬ χ → ¬ φ))) → (((((((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ)) → ¬ φ) → (¬ χ → ¬ φ)) → χ) → (ψχ)))
42, 3ax-mp 8 . . 3 (((((((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ)) → ¬ φ) → (¬ χ → ¬ φ)) → χ) → (ψχ))
5 ax-meredith 1406 . . 3 ((((((((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ)) → ¬ φ) → (¬ χ → ¬ φ)) → χ) → (ψχ)) → (((ψχ) → (((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ))) → (φ → (((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ)))))
64, 5ax-mp 8 . 2 (((ψχ) → (((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ))) → (φ → (((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ))))
71, 6ax-mp 8 1 (φ → (((ψχ) → θ) → (((χτ) → (¬ θ → ¬ ψ)) → θ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4 This theorem was proved from axioms:  ax-mp 8  ax-meredith 1406 This theorem is referenced by:  merlem8  1414
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