 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax2 Structured version   Visualization version   GIF version

Theorem ax2 1589
 Description: Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Proof of Theorem ax2
StepHypRef Expression
1 luklem7 1586 . 2 ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
2 luklem8 1587 . . 3 ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑 → (𝜑𝜒))))
3 luklem6 1585 . . . 4 ((𝜑 → (𝜑𝜒)) → (𝜑𝜒))
4 luklem8 1587 . . . 4 (((𝜑 → (𝜑𝜒)) → (𝜑𝜒)) → (((𝜑𝜓) → (𝜑 → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒))))
53, 4ax-mp 5 . . 3 (((𝜑𝜓) → (𝜑 → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒)))
62, 5luklem1 1580 . 2 ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
71, 6luklem1 1580 1 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → 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: (None)
 Copyright terms: Public domain W3C validator