Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  merlem7 Structured version   Visualization version   GIF version

Theorem merlem7 1564
 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 1561 . 2 ((𝜓𝜒) → (((𝜓𝜒) → 𝜃) → (((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)))
2 merlem6 1563 . . . 4 ((((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃) → (((((𝜓𝜒) → 𝜃) → (((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)) → ¬ 𝜑) → (¬ 𝜒 → ¬ 𝜑)))
3 meredith 1557 . . . 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 5 . . 3 (((((((𝜓𝜒) → 𝜃) → (((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)) → ¬ 𝜑) → (¬ 𝜒 → ¬ 𝜑)) → 𝜒) → (𝜓𝜒))
5 meredith 1557 . . 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 5 . 2 (((𝜓𝜒) → (((𝜓𝜒) → 𝜃) → (((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))) → (𝜑 → (((𝜓𝜒) → 𝜃) → (((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))))
71, 6ax-mp 5 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 5  ax-1 6  ax-2 7  ax-3 8 This theorem is referenced by:  merlem8  1565
 Copyright terms: Public domain W3C validator