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Theorem hirstL-ax3 39512
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) → ((¬ 𝜑𝜓) → 𝜑))

Proof of Theorem hirstL-ax3
StepHypRef Expression
1 pm4.64 385 . 2 ((¬ 𝜑𝜓) ↔ (𝜑𝜓))
2 pm4.66 434 . . 3 ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))
3 pm2.64 825 . . . 4 ((𝜑𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑))
43com12 32 . . 3 ((𝜑 ∨ ¬ 𝜓) → ((𝜑𝜓) → 𝜑))
52, 4sylbi 205 . 2 ((¬ 𝜑 → ¬ 𝜓) → ((𝜑𝜓) → 𝜑))
61, 5syl5bi 230 1 ((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜑𝜓) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381
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 195  df-or 383
This theorem is referenced by:  ax3h  39513
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