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Theorem lukshef-ax2 32874
Description: A single axiom for propositional calculus discovered by Jan Lukasiewicz. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom L2 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.)
Assertion
Ref Expression
lukshef-ax2 ((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜒𝜑)) ⊼ ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))

Proof of Theorem lukshef-ax2
StepHypRef Expression
1 nannan 1615 . . . 4 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
21biimpi 207 . . 3 ((𝜑 ⊼ (𝜓𝜒)) → (𝜑 → (𝜓𝜒)))
3 simpr 477 . . . . 5 ((𝜓𝜒) → 𝜒)
43imim2i 16 . . . 4 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
5 simpl 474 . . . . . 6 ((𝜓𝜒) → 𝜓)
65imim2i 16 . . . . 5 ((𝜑 → (𝜓𝜒)) → (𝜑𝜓))
7 pm2.27 42 . . . . . . 7 (𝜑 → ((𝜑𝜓) → 𝜓))
87anim2d 605 . . . . . 6 (𝜑 → ((𝜃 ∧ (𝜑𝜓)) → (𝜃𝜓)))
98expdimp 444 . . . . 5 ((𝜑𝜃) → ((𝜑𝜓) → (𝜃𝜓)))
106, 9syl5com 31 . . . 4 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜃) → (𝜃𝜓)))
11 ancr 542 . . . . 5 ((𝜑𝜒) → (𝜑 → (𝜒𝜑)))
1211anim1i 608 . . . 4 (((𝜑𝜒) ∧ ((𝜑𝜃) → (𝜃𝜓))) → ((𝜑 → (𝜒𝜑)) ∧ ((𝜑𝜃) → (𝜃𝜓))))
134, 10, 12syl2anc 579 . . 3 ((𝜑 → (𝜓𝜒)) → ((𝜑 → (𝜒𝜑)) ∧ ((𝜑𝜃) → (𝜃𝜓))))
14 con3 150 . . . . 5 (((𝜑𝜃) → (𝜃𝜓)) → (¬ (𝜃𝜓) → ¬ (𝜑𝜃)))
15 df-nan 1609 . . . . 5 ((𝜃𝜓) ↔ ¬ (𝜃𝜓))
16 df-nan 1609 . . . . 5 ((𝜑𝜃) ↔ ¬ (𝜑𝜃))
1714, 15, 163imtr4g 287 . . . 4 (((𝜑𝜃) → (𝜃𝜓)) → ((𝜃𝜓) → (𝜑𝜃)))
1817anim2i 610 . . 3 (((𝜑 → (𝜒𝜑)) ∧ ((𝜑𝜃) → (𝜃𝜓))) → ((𝜑 → (𝜒𝜑)) ∧ ((𝜃𝜓) → (𝜑𝜃))))
19 nannan 1615 . . . . 5 ((𝜑 ⊼ (𝜒𝜑)) ↔ (𝜑 → (𝜒𝜑)))
2019biimpri 219 . . . 4 ((𝜑 → (𝜒𝜑)) → (𝜑 ⊼ (𝜒𝜑)))
21 nanim 1617 . . . . 5 (((𝜃𝜓) → (𝜑𝜃)) ↔ ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))
2221biimpi 207 . . . 4 (((𝜃𝜓) → (𝜑𝜃)) → ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))
2320, 22anim12i 606 . . 3 (((𝜑 → (𝜒𝜑)) ∧ ((𝜃𝜓) → (𝜑𝜃))) → ((𝜑 ⊼ (𝜒𝜑)) ∧ ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
242, 13, 18, 234syl 19 . 2 ((𝜑 ⊼ (𝜓𝜒)) → ((𝜑 ⊼ (𝜒𝜑)) ∧ ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
25 nannan 1615 . 2 (((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜒𝜑)) ⊼ ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))) ↔ ((𝜑 ⊼ (𝜓𝜒)) → ((𝜑 ⊼ (𝜒𝜑)) ∧ ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))))
2624, 25mpbir 222 1 ((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜒𝜑)) ⊼ ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wnan 1608
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 198  df-an 385  df-nan 1609
This theorem is referenced by: (None)
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