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Theorem merlem1 1407
 Description: Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem1 (((χ → (¬ φψ)) → τ) → (φτ))

Proof of Theorem merlem1
StepHypRef Expression
1 ax-meredith 1406 . . 3 (((((¬ φψ) → (¬ (¬ τ → ¬ χ) → ¬ ¬ (¬ φψ))) → (¬ τ → ¬ χ)) → τ) → ((τ → ¬ φ) → (¬ (¬ φψ) → ¬ φ)))
2 ax-meredith 1406 . . 3 ((((((¬ φψ) → (¬ (¬ τ → ¬ χ) → ¬ ¬ (¬ φψ))) → (¬ τ → ¬ χ)) → τ) → ((τ → ¬ φ) → (¬ (¬ φψ) → ¬ φ))) → ((((τ → ¬ φ) → (¬ (¬ φψ) → ¬ φ)) → (¬ φψ)) → (χ → (¬ φψ))))
31, 2ax-mp 8 . 2 ((((τ → ¬ φ) → (¬ (¬ φψ) → ¬ φ)) → (¬ φψ)) → (χ → (¬ φψ)))
4 ax-meredith 1406 . 2 (((((τ → ¬ φ) → (¬ (¬ φψ) → ¬ φ)) → (¬ φψ)) → (χ → (¬ φψ))) → (((χ → (¬ φψ)) → τ) → (φτ)))
53, 4ax-mp 8 1 (((χ → (¬ φψ)) → τ) → (φτ))
 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:  merlem2  1408  merlem5  1411  luk-3  1422
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