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Theorem ax2 1737
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 1734 . 2 ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
2 luklem8 1735 . . 3 ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑 → (𝜑𝜒))))
3 luklem6 1733 . . . 4 ((𝜑 → (𝜑𝜒)) → (𝜑𝜒))
4 luklem8 1735 . . . 4 (((𝜑 → (𝜑𝜒)) → (𝜑𝜒)) → (((𝜑𝜓) → (𝜑 → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒))))
53, 4ax-mp 5 . . 3 (((𝜑𝜓) → (𝜑 → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒)))
62, 5luklem1 1728 . 2 ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
71, 6luklem1 1728 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)
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