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Theorem meran3 32089
Description: A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
Assertion
Ref Expression
meran3 (¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜒𝜑) ∨ (𝜏 ∨ (𝜃𝜑))))

Proof of Theorem meran3
StepHypRef Expression
1 pm2.3 595 . . . . . 6 ((𝜒 ∨ (𝜃𝜏)) → (𝜒 ∨ (𝜏𝜃)))
21imim2i 16 . . . . 5 (((¬ 𝜑𝜓) → (𝜒 ∨ (𝜃𝜏))) → ((¬ 𝜑𝜓) → (𝜒 ∨ (𝜏𝜃))))
3 pm1.5 544 . . . . 5 ((𝜒 ∨ (𝜏𝜃)) → (𝜏 ∨ (𝜒𝜃)))
42, 3syl6 35 . . . 4 (((¬ 𝜑𝜓) → (𝜒 ∨ (𝜃𝜏))) → ((¬ 𝜑𝜓) → (𝜏 ∨ (𝜒𝜃))))
5 imor 428 . . . 4 (((¬ 𝜑𝜓) → (𝜒 ∨ (𝜃𝜏))) ↔ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))))
6 imor 428 . . . 4 (((¬ 𝜑𝜓) → (𝜏 ∨ (𝜒𝜃))) ↔ (¬ (¬ 𝜑𝜓) ∨ (𝜏 ∨ (𝜒𝜃))))
74, 5, 63imtr3i 280 . . 3 ((¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) → (¬ (¬ 𝜑𝜓) ∨ (𝜏 ∨ (𝜒𝜃))))
8 meran1 32087 . . . 4 (¬ (¬ (¬ 𝜑𝜓) ∨ (𝜏 ∨ (𝜒𝜃))) ∨ (¬ (¬ 𝜒𝜑) ∨ (𝜏 ∨ (𝜃𝜑))))
98imorri 430 . . 3 ((¬ (¬ 𝜑𝜓) ∨ (𝜏 ∨ (𝜒𝜃))) → (¬ (¬ 𝜒𝜑) ∨ (𝜏 ∨ (𝜃𝜑))))
107, 9syl 17 . 2 ((¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) → (¬ (¬ 𝜒𝜑) ∨ (𝜏 ∨ (𝜃𝜑))))
1110imori 429 1 (¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜒𝜑) ∨ (𝜏 ∨ (𝜃𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383
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 197  df-or 385
This theorem is referenced by: (None)
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