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Theorem meran1 31444
Description: A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
Assertion
Ref Expression
meran1 (¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))

Proof of Theorem meran1
StepHypRef Expression
1 orc 398 . . . . . 6 𝜑 → (¬ 𝜑𝜓))
2 olc 397 . . . . . 6 (𝜓 → (¬ 𝜑𝜓))
31, 2ja 171 . . . . 5 ((𝜑𝜓) → (¬ 𝜑𝜓))
43imim1i 60 . . . 4 (((¬ 𝜑𝜓) → (𝜒 ∨ (𝜃𝜏))) → ((𝜑𝜓) → (𝜒 ∨ (𝜃𝜏))))
5 pm2.24 119 . . . . . 6 (𝜃 → (¬ 𝜃𝜑))
6 idd 24 . . . . . 6 (𝜃 → (𝜑𝜑))
75, 6jaod 393 . . . . 5 (𝜃 → ((¬ 𝜃𝜑) → 𝜑))
87com12 32 . . . 4 ((¬ 𝜃𝜑) → (𝜃𝜑))
9 pm1.5 542 . . . . . 6 ((¬ (𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) → (𝜒 ∨ (¬ (𝜑𝜓) ∨ (𝜃𝜏))))
10 pm2.3 593 . . . . . . . 8 ((¬ (𝜑𝜓) ∨ (𝜃𝜏)) → (¬ (𝜑𝜓) ∨ (𝜏𝜃)))
11 pm1.5 542 . . . . . . . 8 ((¬ (𝜑𝜓) ∨ (𝜏𝜃)) → (𝜏 ∨ (¬ (𝜑𝜓) ∨ 𝜃)))
12 pm2.21 118 . . . . . . . . . . . . 13 𝜑 → (𝜑𝜓))
13 mth8 156 . . . . . . . . . . . . 13 (𝜃 → (¬ 𝜑 → ¬ (𝜃𝜑)))
1412, 13imim12i 59 . . . . . . . . . . . 12 (((𝜑𝜓) → 𝜃) → (¬ 𝜑 → (¬ 𝜑 → ¬ (𝜃𝜑))))
1514pm2.43d 50 . . . . . . . . . . 11 (((𝜑𝜓) → 𝜃) → (¬ 𝜑 → ¬ (𝜃𝜑)))
1615con4d 112 . . . . . . . . . 10 (((𝜑𝜓) → 𝜃) → ((𝜃𝜑) → 𝜑))
17 imor 426 . . . . . . . . . 10 (((𝜑𝜓) → 𝜃) ↔ (¬ (𝜑𝜓) ∨ 𝜃))
18 imor 426 . . . . . . . . . 10 (((𝜃𝜑) → 𝜑) ↔ (¬ (𝜃𝜑) ∨ 𝜑))
1916, 17, 183imtr3i 278 . . . . . . . . 9 ((¬ (𝜑𝜓) ∨ 𝜃) → (¬ (𝜃𝜑) ∨ 𝜑))
2019orim2i 538 . . . . . . . 8 ((𝜏 ∨ (¬ (𝜑𝜓) ∨ 𝜃)) → (𝜏 ∨ (¬ (𝜃𝜑) ∨ 𝜑)))
21 pm1.5 542 . . . . . . . 8 ((𝜏 ∨ (¬ (𝜃𝜑) ∨ 𝜑)) → (¬ (𝜃𝜑) ∨ (𝜏𝜑)))
2210, 11, 20, 214syl 19 . . . . . . 7 ((¬ (𝜑𝜓) ∨ (𝜃𝜏)) → (¬ (𝜃𝜑) ∨ (𝜏𝜑)))
2322orim2i 538 . . . . . 6 ((𝜒 ∨ (¬ (𝜑𝜓) ∨ (𝜃𝜏))) → (𝜒 ∨ (¬ (𝜃𝜑) ∨ (𝜏𝜑))))
24 pm1.5 542 . . . . . 6 ((𝜒 ∨ (¬ (𝜃𝜑) ∨ (𝜏𝜑))) → (¬ (𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))
259, 23, 243syl 18 . . . . 5 ((¬ (𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) → (¬ (𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))
26 imor 426 . . . . 5 (((𝜑𝜓) → (𝜒 ∨ (𝜃𝜏))) ↔ (¬ (𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))))
27 imor 426 . . . . 5 (((𝜃𝜑) → (𝜒 ∨ (𝜏𝜑))) ↔ (¬ (𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))
2825, 26, 273imtr4i 279 . . . 4 (((𝜑𝜓) → (𝜒 ∨ (𝜃𝜏))) → ((𝜃𝜑) → (𝜒 ∨ (𝜏𝜑))))
294, 8, 28syl2im 39 . . 3 (((¬ 𝜑𝜓) → (𝜒 ∨ (𝜃𝜏))) → ((¬ 𝜃𝜑) → (𝜒 ∨ (𝜏𝜑))))
30 imor 426 . . 3 (((¬ 𝜑𝜓) → (𝜒 ∨ (𝜃𝜏))) ↔ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))))
31 imor 426 . . 3 (((¬ 𝜃𝜑) → (𝜒 ∨ (𝜏𝜑))) ↔ (¬ (¬ 𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))
3229, 30, 313imtr3i 278 . 2 ((¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) → (¬ (¬ 𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))
3332imori 427 1 (¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))
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:  meran2  31445  meran3  31446
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