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Theorem axunnd 9995
Description: A version of the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2391. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
Assertion
Ref Expression
axunnd 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)

Proof of Theorem axunnd
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 axunndlem1 9994 . . . 4 𝑤𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤)
2 nfnae 2457 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3 nfnae 2457 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
42, 3nfan 1901 . . . . 5 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
5 nfnae 2457 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
6 nfnae 2457 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑧
75, 6nfan 1901 . . . . . 6 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
8 nfv 1916 . . . . . . . 8 𝑤(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
9 nfcvf 3001 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
109adantr 484 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑦)
11 nfcvd 2975 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑤)
1210, 11nfeld 2985 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦𝑤)
13 nfcvf 3001 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑧)
1413adantl 485 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑧)
1511, 14nfeld 2985 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤𝑧)
1612, 15nfand 1899 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑦𝑤𝑤𝑧))
178, 16nfexd 2349 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑤(𝑦𝑤𝑤𝑧))
1817, 12nfimd 1896 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤))
197, 18nfald 2348 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤))
20 nfcvd 2975 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑦𝑤)
21 nfcvf2 3002 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
2221adantr 484 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑦𝑥)
2320, 22nfeqd 2984 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑦 𝑤 = 𝑥)
247, 23nfan1 2201 . . . . . . 7 𝑦((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
25 elequ2 2130 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
26 elequ1 2122 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
2725, 26anbi12d 633 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((𝑦𝑤𝑤𝑧) ↔ (𝑦𝑥𝑥𝑧)))
2827a1i 11 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑦𝑤𝑤𝑧) ↔ (𝑦𝑥𝑥𝑧))))
294, 16, 28cbvexd 2430 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤(𝑦𝑤𝑤𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
3029adantr 484 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑤(𝑦𝑤𝑤𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
3125adantl 485 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑦𝑤𝑦𝑥))
3230, 31imbi12d 348 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
3324, 32albid 2225 . . . . . 6 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
3433ex 416 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → (∀𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))))
354, 19, 34cbvexd 2430 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
361, 35mpbii 236 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
3736ex 416 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
38 nfae 2456 . . . 4 𝑦𝑥 𝑥 = 𝑦
39 nfae 2456 . . . . . 6 𝑥𝑥 𝑥 = 𝑦
40 elirrv 9036 . . . . . . . . 9 ¬ 𝑦𝑦
41 elequ2 2130 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
4240, 41mtbiri 330 . . . . . . . 8 (𝑥 = 𝑦 → ¬ 𝑦𝑥)
4342intnanrd 493 . . . . . . 7 (𝑥 = 𝑦 → ¬ (𝑦𝑥𝑥𝑧))
4443sps 2185 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ¬ (𝑦𝑥𝑥𝑧))
4539, 44nexd 2224 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
4645pm2.21d 121 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
4738, 46alrimi 2214 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
484719.8ad 2182 . 2 (∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
49 nfae 2456 . . . 4 𝑦𝑥 𝑥 = 𝑧
50 nfae 2456 . . . . . 6 𝑥𝑥 𝑥 = 𝑧
51 elirrv 9036 . . . . . . . . 9 ¬ 𝑧𝑧
52 elequ1 2122 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝑧𝑧𝑧))
5351, 52mtbiri 330 . . . . . . . 8 (𝑥 = 𝑧 → ¬ 𝑥𝑧)
5453intnand 492 . . . . . . 7 (𝑥 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
5554sps 2185 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
5650, 55nexd 2224 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
5756pm2.21d 121 . . . 4 (∀𝑥 𝑥 = 𝑧 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
5849, 57alrimi 2214 . . 3 (∀𝑥 𝑥 = 𝑧 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
595819.8ad 2182 . 2 (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
6037, 48, 59pm2.61ii 186 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536  wex 1781  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-13 2391  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-un 7436  ax-reg 9032
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-eprel 5438  df-fr 5487
This theorem is referenced by:  zfcndun  10014  axunprim  32937
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