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Theorem rp-fakenanass 38177
Description: A special case where nand appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakenanass ((𝜑𝜒) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))

Proof of Theorem rp-fakenanass
StepHypRef Expression
1 bicom1 211 . . . 4 ((𝜑𝜒) → (𝜒𝜑))
2 nanbi2 1496 . . . 4 ((𝜑𝜒) → ((𝜓𝜑) ↔ (𝜓𝜒)))
31, 2nanbi12d 1503 . . 3 ((𝜑𝜒) → ((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜑 ⊼ (𝜓𝜒))))
4 nannan 1491 . . . . . 6 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
5 simpr 476 . . . . . . 7 ((𝜓𝜒) → 𝜒)
65imim2i 16 . . . . . 6 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
74, 6sylbi 207 . . . . 5 ((𝜑 ⊼ (𝜓𝜒)) → (𝜑𝜒))
8 nannan 1491 . . . . . 6 ((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜒 → (𝜓𝜑)))
9 simpr 476 . . . . . . 7 ((𝜓𝜑) → 𝜑)
109imim2i 16 . . . . . 6 ((𝜒 → (𝜓𝜑)) → (𝜒𝜑))
118, 10sylbi 207 . . . . 5 ((𝜒 ⊼ (𝜓𝜑)) → (𝜒𝜑))
127, 11impbid21d 201 . . . 4 ((𝜒 ⊼ (𝜓𝜑)) → ((𝜑 ⊼ (𝜓𝜒)) → (𝜑𝜒)))
138notbii 309 . . . . . . 7 (¬ (𝜒 ⊼ (𝜓𝜑)) ↔ ¬ (𝜒 → (𝜓𝜑)))
14 pm4.61 441 . . . . . . 7 (¬ (𝜒 → (𝜓𝜑)) ↔ (𝜒 ∧ ¬ (𝜓𝜑)))
15 ianor 508 . . . . . . . 8 (¬ (𝜓𝜑) ↔ (¬ 𝜓 ∨ ¬ 𝜑))
1615anbi2i 730 . . . . . . 7 ((𝜒 ∧ ¬ (𝜓𝜑)) ↔ (𝜒 ∧ (¬ 𝜓 ∨ ¬ 𝜑)))
1713, 14, 163bitri 286 . . . . . 6 (¬ (𝜒 ⊼ (𝜓𝜑)) ↔ (𝜒 ∧ (¬ 𝜓 ∨ ¬ 𝜑)))
184notbii 309 . . . . . . 7 (¬ (𝜑 ⊼ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒)))
19 pm4.61 441 . . . . . . 7 (¬ (𝜑 → (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒)))
20 ianor 508 . . . . . . . 8 (¬ (𝜓𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒))
2120anbi2i 730 . . . . . . 7 ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ (𝜑 ∧ (¬ 𝜓 ∨ ¬ 𝜒)))
2218, 19, 213bitri 286 . . . . . 6 (¬ (𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 ∧ (¬ 𝜓 ∨ ¬ 𝜒)))
23 pm5.1 920 . . . . . . . 8 ((𝜑𝜒) → (𝜑𝜒))
2423ancoms 468 . . . . . . 7 ((𝜒𝜑) → (𝜑𝜒))
2524ad2ant2r 798 . . . . . 6 (((𝜒 ∧ (¬ 𝜓 ∨ ¬ 𝜑)) ∧ (𝜑 ∧ (¬ 𝜓 ∨ ¬ 𝜒))) → (𝜑𝜒))
2617, 22, 25syl2anb 495 . . . . 5 ((¬ (𝜒 ⊼ (𝜓𝜑)) ∧ ¬ (𝜑 ⊼ (𝜓𝜒))) → (𝜑𝜒))
2726ex 449 . . . 4 (¬ (𝜒 ⊼ (𝜓𝜑)) → (¬ (𝜑 ⊼ (𝜓𝜒)) → (𝜑𝜒)))
2812, 27bija 369 . . 3 (((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜑 ⊼ (𝜓𝜒))) → (𝜑𝜒))
293, 28impbii 199 . 2 ((𝜑𝜒) ↔ ((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜑 ⊼ (𝜓𝜒))))
30 nancom 1490 . . . . 5 ((𝜓𝜑) ↔ (𝜑𝜓))
3130nanbi2i 1499 . . . 4 ((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜒 ⊼ (𝜑𝜓)))
32 nancom 1490 . . . 4 ((𝜒 ⊼ (𝜑𝜓)) ↔ ((𝜑𝜓) ⊼ 𝜒))
3331, 32bitri 264 . . 3 ((𝜒 ⊼ (𝜓𝜑)) ↔ ((𝜑𝜓) ⊼ 𝜒))
3433bibi1i 327 . 2 (((𝜒 ⊼ (𝜓𝜑)) ↔ (𝜑 ⊼ (𝜓𝜒))) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))
3529, 34bitri 264 1 ((𝜑𝜒) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wnan 1487
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 197  df-or 384  df-an 385  df-nan 1488
This theorem is referenced by: (None)
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