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Theorem waj-ax 32921
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 1616 . . 3 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
2 simpr 478 . . . . . . . . 9 ((𝜓𝜒) → 𝜒)
32imim2i 16 . . . . . . . 8 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
4 pm2.27 42 . . . . . . . . . 10 (𝜑 → ((𝜑𝜒) → 𝜒))
54anim2d 606 . . . . . . . . 9 (𝜑 → ((𝜃 ∧ (𝜑𝜒)) → (𝜃𝜒)))
65expdimp 445 . . . . . . . 8 ((𝜑𝜃) → ((𝜑𝜒) → (𝜃𝜒)))
73, 6syl5com 31 . . . . . . 7 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜃) → (𝜃𝜒)))
87con3d 150 . . . . . 6 ((𝜑 → (𝜓𝜒)) → (¬ (𝜃𝜒) → ¬ (𝜑𝜃)))
9 df-nan 1610 . . . . . 6 ((𝜃𝜒) ↔ ¬ (𝜃𝜒))
10 df-nan 1610 . . . . . 6 ((𝜑𝜃) ↔ ¬ (𝜑𝜃))
118, 9, 103imtr4g 288 . . . . 5 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) → (𝜑𝜃)))
12 nanim 1618 . . . . 5 (((𝜃𝜒) → (𝜑𝜃)) ↔ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))
1311, 12sylib 210 . . . 4 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))
14 pm3.21 464 . . . . . . . 8 (𝜓 → (𝜑 → (𝜑𝜓)))
1514adantr 473 . . . . . . 7 ((𝜓𝜒) → (𝜑 → (𝜑𝜓)))
1615com12 32 . . . . . 6 (𝜑 → ((𝜓𝜒) → (𝜑𝜓)))
1716a2i 14 . . . . 5 ((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜑𝜓)))
18 nannan 1616 . . . . 5 ((𝜑 ⊼ (𝜑𝜓)) ↔ (𝜑 → (𝜑𝜓)))
1917, 18sylibr 226 . . . 4 ((𝜑 → (𝜓𝜒)) → (𝜑 ⊼ (𝜑𝜓)))
2013, 19jca 508 . . 3 ((𝜑 → (𝜓𝜒)) → (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ∧ (𝜑 ⊼ (𝜑𝜓))))
211, 20sylbi 209 . 2 ((𝜑 ⊼ (𝜓𝜒)) → (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ∧ (𝜑 ⊼ (𝜑𝜓))))
22 nannan 1616 . 2 (((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓)))) ↔ ((𝜑 ⊼ (𝜓𝜒)) → (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ∧ (𝜑 ⊼ (𝜑𝜓)))))
2321, 22mpbir 223 1 ((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wnan 1609
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 199  df-an 386  df-nan 1610
This theorem is referenced by: (None)
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