 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  merlem1 Structured version   Visualization version   GIF version

Theorem merlem1 1715
 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 meredith 1714 . . 3 (((((¬ 𝜑𝜓) → (¬ (¬ 𝜏 → ¬ 𝜒) → ¬ ¬ (¬ 𝜑𝜓))) → (¬ 𝜏 → ¬ 𝜒)) → 𝜏) → ((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑𝜓) → ¬ 𝜑)))
2 meredith 1714 . . 3 ((((((¬ 𝜑𝜓) → (¬ (¬ 𝜏 → ¬ 𝜒) → ¬ ¬ (¬ 𝜑𝜓))) → (¬ 𝜏 → ¬ 𝜒)) → 𝜏) → ((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑𝜓) → ¬ 𝜑))) → ((((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑𝜓) → ¬ 𝜑)) → (¬ 𝜑𝜓)) → (𝜒 → (¬ 𝜑𝜓))))
31, 2ax-mp 5 . 2 ((((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑𝜓) → ¬ 𝜑)) → (¬ 𝜑𝜓)) → (𝜒 → (¬ 𝜑𝜓)))
4 meredith 1714 . 2 (((((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑𝜓) → ¬ 𝜑)) → (¬ 𝜑𝜓)) → (𝜒 → (¬ 𝜑𝜓))) → (((𝜒 → (¬ 𝜑𝜓)) → 𝜏) → (𝜑𝜏)))
53, 4ax-mp 5 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 5  ax-1 6  ax-2 7  ax-3 8 This theorem is referenced by:  merlem2  1716  merlem5  1719  luk-3  1730
 Copyright terms: Public domain W3C validator