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Theorem axacprim 31710
Description: ax-ac 9319 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.)
Assertion
Ref Expression
axacprim ¬ ∀𝑥 ¬ ∀𝑦𝑧(∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))

Proof of Theorem axacprim
StepHypRef Expression
1 axacnd 9472 . 2 𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
2 df-an 385 . . . . . . 7 ((𝑦𝑧𝑧𝑤) ↔ ¬ (𝑦𝑧 → ¬ 𝑧𝑤))
32albii 1787 . . . . . 6 (∀𝑥(𝑦𝑧𝑧𝑤) ↔ ∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤))
4 anass 682 . . . . . . . . . . . . . 14 (((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ (𝑦𝑧 ∧ (𝑧𝑤 ∧ (𝑦𝑤𝑤𝑥))))
5 annim 440 . . . . . . . . . . . . . . . 16 ((𝑧𝑤 ∧ ¬ (𝑦𝑤 → ¬ 𝑤𝑥)) ↔ ¬ (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)))
6 pm4.63 436 . . . . . . . . . . . . . . . . 17 (¬ (𝑦𝑤 → ¬ 𝑤𝑥) ↔ (𝑦𝑤𝑤𝑥))
76anbi2i 730 . . . . . . . . . . . . . . . 16 ((𝑧𝑤 ∧ ¬ (𝑦𝑤 → ¬ 𝑤𝑥)) ↔ (𝑧𝑤 ∧ (𝑦𝑤𝑤𝑥)))
85, 7bitr3i 266 . . . . . . . . . . . . . . 15 (¬ (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)) ↔ (𝑧𝑤 ∧ (𝑦𝑤𝑤𝑥)))
98anbi2i 730 . . . . . . . . . . . . . 14 ((𝑦𝑧 ∧ ¬ (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ (𝑦𝑧 ∧ (𝑧𝑤 ∧ (𝑦𝑤𝑤𝑥))))
10 annim 440 . . . . . . . . . . . . . 14 ((𝑦𝑧 ∧ ¬ (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ ¬ (𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
114, 9, 103bitr2i 288 . . . . . . . . . . . . 13 (((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ¬ (𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
1211exbii 1814 . . . . . . . . . . . 12 (∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ∃𝑤 ¬ (𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
13 exnal 1794 . . . . . . . . . . . 12 (∃𝑤 ¬ (𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
1412, 13bitri 264 . . . . . . . . . . 11 (∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
1514bibi1i 327 . . . . . . . . . 10 ((∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ (¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ 𝑦 = 𝑤))
16 dfbi1 203 . . . . . . . . . 10 ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ 𝑦 = 𝑤) ↔ ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
1715, 16bitri 264 . . . . . . . . 9 ((∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
1817albii 1787 . . . . . . . 8 (∀𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
1918exbii 1814 . . . . . . 7 (∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ∃𝑤𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
20 df-ex 1745 . . . . . . 7 (∃𝑤𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))) ↔ ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
2119, 20bitri 264 . . . . . 6 (∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
223, 21imbi12i 339 . . . . 5 ((∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ (∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)))))))
23222albii 1788 . . . 4 (∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∀𝑦𝑧(∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)))))))
2423exbii 1814 . . 3 (∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∃𝑥𝑦𝑧(∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)))))))
25 df-ex 1745 . . 3 (∃𝑥𝑦𝑧(∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)))))) ↔ ¬ ∀𝑥 ¬ ∀𝑦𝑧(∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)))))))
2624, 25bitri 264 . 2 (∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ ¬ ∀𝑥 ¬ ∀𝑦𝑧(∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)))))))
271, 26mpbi 220 1 ¬ ∀𝑥 ¬ ∀𝑦𝑧(∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538  ax-ac 9319
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-eprel 5058  df-fr 5102
This theorem is referenced by: (None)
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