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Theorem axacprim 34318
Description: ax-ac 10402 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 10555 . 2 𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
2 df-an 398 . . . . . . 7 ((𝑦𝑧𝑧𝑤) ↔ ¬ (𝑦𝑧 → ¬ 𝑧𝑤))
32albii 1822 . . . . . 6 (∀𝑥(𝑦𝑧𝑧𝑤) ↔ ∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤))
4 anass 470 . . . . . . . . . . . . . 14 (((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ (𝑦𝑧 ∧ (𝑧𝑤 ∧ (𝑦𝑤𝑤𝑥))))
5 annim 405 . . . . . . . . . . . . . . . 16 ((𝑧𝑤 ∧ ¬ (𝑦𝑤 → ¬ 𝑤𝑥)) ↔ ¬ (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)))
6 pm4.63 399 . . . . . . . . . . . . . . . . 17 (¬ (𝑦𝑤 → ¬ 𝑤𝑥) ↔ (𝑦𝑤𝑤𝑥))
76anbi2i 624 . . . . . . . . . . . . . . . 16 ((𝑧𝑤 ∧ ¬ (𝑦𝑤 → ¬ 𝑤𝑥)) ↔ (𝑧𝑤 ∧ (𝑦𝑤𝑤𝑥)))
85, 7bitr3i 277 . . . . . . . . . . . . . . 15 (¬ (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥)) ↔ (𝑧𝑤 ∧ (𝑦𝑤𝑤𝑥)))
98anbi2i 624 . . . . . . . . . . . . . 14 ((𝑦𝑧 ∧ ¬ (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ (𝑦𝑧 ∧ (𝑧𝑤 ∧ (𝑦𝑤𝑤𝑥))))
10 annim 405 . . . . . . . . . . . . . 14 ((𝑦𝑧 ∧ ¬ (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ ¬ (𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
114, 9, 103bitr2i 299 . . . . . . . . . . . . 13 (((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ¬ (𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
1211exbii 1851 . . . . . . . . . . . 12 (∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ∃𝑤 ¬ (𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
13 exnal 1830 . . . . . . . . . . . 12 (∃𝑤 ¬ (𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
1412, 13bitri 275 . . . . . . . . . . 11 (∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))
1514bibi1i 339 . . . . . . . . . 10 ((∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ (¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ 𝑦 = 𝑤))
16 dfbi1 212 . . . . . . . . . 10 ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) ↔ 𝑦 = 𝑤) ↔ ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
1715, 16bitri 275 . . . . . . . . 9 ((∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
1817albii 1822 . . . . . . . 8 (∀𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
1918exbii 1851 . . . . . . 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 1783 . . . . . . 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 275 . . . . . 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 351 . . . . 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 1823 . . . 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 1851 . . 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 1783 . . 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 275 . 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 229 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 205  wa 397  wal 1540  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-reg 9535  ax-ac 10402
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-eprel 5542  df-fr 5593
This theorem is referenced by: (None)
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