Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  merlem12 Structured version   Visualization version   GIF version

Theorem merlem12 1569
 Description: Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem12 (((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → 𝜑)

Proof of Theorem merlem12
StepHypRef Expression
1 merlem5 1562 . . . 4 ((𝜒𝜒) → (¬ ¬ 𝜒𝜒))
2 merlem2 1559 . . . 4 (((𝜒𝜒) → (¬ ¬ 𝜒𝜒)) → (𝜃 → (¬ ¬ 𝜒𝜒)))
31, 2ax-mp 5 . . 3 (𝜃 → (¬ ¬ 𝜒𝜒))
4 merlem4 1561 . . 3 ((𝜃 → (¬ ¬ 𝜒𝜒)) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → 𝜑)))
53, 4ax-mp 5 . 2 (((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → 𝜑))
6 merlem11 1568 . 2 ((((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → 𝜑)) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → 𝜑))
75, 6ax-mp 5 1 (((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → 𝜑)
 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:  merlem13  1570
 Copyright terms: Public domain W3C validator