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Theorem waj-ax 33836
 Description: A single axiom for propositional calculus discovered by Mordchaj Wajsberg (Logical Works, Polish Academy of Sciences, 1977). See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom W on slide 8). (Contributed by Anthony Hart, 13-Aug-2011.)
Assertion
Ref Expression
waj-ax ((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓))))

Proof of Theorem waj-ax
StepHypRef Expression
1 nannan 1488 . . 3 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
2 simpr 488 . . . . . . . . 9 ((𝜓𝜒) → 𝜒)
32imim2i 16 . . . . . . . 8 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
4 pm2.27 42 . . . . . . . . . 10 (𝜑 → ((𝜑𝜒) → 𝜒))
54anim2d 614 . . . . . . . . 9 (𝜑 → ((𝜃 ∧ (𝜑𝜒)) → (𝜃𝜒)))
65expdimp 456 . . . . . . . 8 ((𝜑𝜃) → ((𝜑𝜒) → (𝜃𝜒)))
73, 6syl5com 31 . . . . . . 7 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜃) → (𝜃𝜒)))
87con3d 155 . . . . . 6 ((𝜑 → (𝜓𝜒)) → (¬ (𝜃𝜒) → ¬ (𝜑𝜃)))
9 df-nan 1483 . . . . . 6 ((𝜃𝜒) ↔ ¬ (𝜃𝜒))
10 df-nan 1483 . . . . . 6 ((𝜑𝜃) ↔ ¬ (𝜑𝜃))
118, 9, 103imtr4g 299 . . . . 5 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) → (𝜑𝜃)))
12 nanim 1489 . . . . 5 (((𝜃𝜒) → (𝜑𝜃)) ↔ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))
1311, 12sylib 221 . . . 4 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))
14 pm3.21 475 . . . . . . . 8 (𝜓 → (𝜑 → (𝜑𝜓)))
1514adantr 484 . . . . . . 7 ((𝜓𝜒) → (𝜑 → (𝜑𝜓)))
1615com12 32 . . . . . 6 (𝜑 → ((𝜓𝜒) → (𝜑𝜓)))
1716a2i 14 . . . . 5 ((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜑𝜓)))
18 nannan 1488 . . . . 5 ((𝜑 ⊼ (𝜑𝜓)) ↔ (𝜑 → (𝜑𝜓)))
1917, 18sylibr 237 . . . 4 ((𝜑 → (𝜓𝜒)) → (𝜑 ⊼ (𝜑𝜓)))
2013, 19jca 515 . . 3 ((𝜑 → (𝜓𝜒)) → (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ∧ (𝜑 ⊼ (𝜑𝜓))))
211, 20sylbi 220 . 2 ((𝜑 ⊼ (𝜓𝜒)) → (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ∧ (𝜑 ⊼ (𝜑𝜓))))
22 nannan 1488 . 2 (((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓)))) ↔ ((𝜑 ⊼ (𝜓𝜒)) → (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ∧ (𝜑 ⊼ (𝜑𝜓)))))
2321, 22mpbir 234 1 ((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ⊼ wnan 1482 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 210  df-an 400  df-nan 1483 This theorem is referenced by: (None)
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