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Theorem nanass 1505
Description: A characterization of when an expression involving alternative denials associates. Remark: alternative denial is commutative, see nancom 1491. (Contributed by Richard Penner, 29-Feb-2020.) (Proof shortened by Wolf Lammen, 23-Oct-2022.)
Assertion
Ref Expression
nanass ((𝜑𝜒) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))

Proof of Theorem nanass
StepHypRef Expression
1 bicom1 220 . . . 4 ((𝜑𝜒) → (𝜒𝜑))
2 nanbi2 1497 . . . 4 ((𝜑𝜒) → ((𝜓𝜑) ↔ (𝜓𝜒)))
31, 2nanbi12d 1504 . . 3 ((𝜑𝜒) → ((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜑 ⊼ (𝜓𝜒))))
4 nannan 1492 . . . . . 6 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
5 simpr 485 . . . . . . 7 ((𝜓𝜒) → 𝜒)
65imim2i 16 . . . . . 6 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
74, 6sylbi 216 . . . . 5 ((𝜑 ⊼ (𝜓𝜒)) → (𝜑𝜒))
8 nannan 1492 . . . . . 6 ((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜒 → (𝜓𝜑)))
9 simpr 485 . . . . . . 7 ((𝜓𝜑) → 𝜑)
109imim2i 16 . . . . . 6 ((𝜒 → (𝜓𝜑)) → (𝜒𝜑))
118, 10sylbi 216 . . . . 5 ((𝜒 ⊼ (𝜓𝜑)) → (𝜒𝜑))
127, 11impbid21d 210 . . . 4 ((𝜒 ⊼ (𝜓𝜑)) → ((𝜑 ⊼ (𝜓𝜒)) → (𝜑𝜒)))
13 nanan 1488 . . . . . 6 ((𝜑 ∧ (𝜓𝜒)) ↔ ¬ (𝜑 ⊼ (𝜓𝜒)))
14 simpl 483 . . . . . 6 ((𝜑 ∧ (𝜓𝜒)) → 𝜑)
1513, 14sylbir 234 . . . . 5 (¬ (𝜑 ⊼ (𝜓𝜒)) → 𝜑)
16 nanan 1488 . . . . . 6 ((𝜒 ∧ (𝜓𝜑)) ↔ ¬ (𝜒 ⊼ (𝜓𝜑)))
17 simpl 483 . . . . . 6 ((𝜒 ∧ (𝜓𝜑)) → 𝜒)
1816, 17sylbir 234 . . . . 5 (¬ (𝜒 ⊼ (𝜓𝜑)) → 𝜒)
19 pm5.1im 262 . . . . 5 (𝜑 → (𝜒 → (𝜑𝜒)))
2015, 18, 19syl2imc 41 . . . 4 (¬ (𝜒 ⊼ (𝜓𝜑)) → (¬ (𝜑 ⊼ (𝜓𝜒)) → (𝜑𝜒)))
2112, 20bija 382 . . 3 (((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜑 ⊼ (𝜓𝜒))) → (𝜑𝜒))
223, 21impbii 208 . 2 ((𝜑𝜒) ↔ ((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜑 ⊼ (𝜓𝜒))))
23 nancom 1491 . . . . 5 ((𝜓𝜑) ↔ (𝜑𝜓))
2423nanbi2i 1500 . . . 4 ((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜒 ⊼ (𝜑𝜓)))
25 nancom 1491 . . . 4 ((𝜒 ⊼ (𝜑𝜓)) ↔ ((𝜑𝜓) ⊼ 𝜒))
2624, 25bitri 274 . . 3 ((𝜒 ⊼ (𝜓𝜑)) ↔ ((𝜑𝜓) ⊼ 𝜒))
2726bibi1i 339 . 2 (((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜑 ⊼ (𝜓𝜒))) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))
2822, 27bitri 274 1 ((𝜑𝜒) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wnan 1486
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 206  df-an 397  df-nan 1487
This theorem is referenced by: (None)
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