Theorem hirstL-ax3 44387

Description: The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.)

Assertion

Ref	Expression
hirstL-ax3	⊢ ((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜑 → 𝜓) → 𝜑))

Proof of Theorem hirstL-ax3

Step	Hyp	Ref	Expression
1		pm4.64 846	. 2 ⊢ ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓))
2		pm4.66 847	. . 3 ⊢ ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))
3		pm2.64 939	. . . 4 ⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑))
4	3	com12 32	. . 3 ⊢ ((𝜑 ∨ ¬ 𝜓) → ((𝜑 ∨ 𝜓) → 𝜑))
5	2, 4	sylbi 216	. 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → ((𝜑 ∨ 𝜓) → 𝜑))
6	1, 5	syl5bi 241	1 ⊢ ((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜑 → 𝜓) → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-or 845

This theorem is referenced by:  ax3h  44388