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Theorem axunnd 10621
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 2365. (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 10620 . . . 4 𝑤𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤)
2 nfnae 2427 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3 nfnae 2427 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
42, 3nfan 1894 . . . . 5 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
5 nfnae 2427 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
6 nfnae 2427 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑧
75, 6nfan 1894 . . . . . 6 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
8 nfv 1909 . . . . . . . 8 𝑤(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
9 nfcvf 2921 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
109adantr 479 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑦)
11 nfcvd 2892 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑤)
1210, 11nfeld 2903 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦𝑤)
13 nfcvf 2921 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑧)
1413adantl 480 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑧)
1511, 14nfeld 2903 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤𝑧)
1612, 15nfand 1892 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑦𝑤𝑤𝑧))
178, 16nfexd 2317 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑤(𝑦𝑤𝑤𝑧))
1817, 12nfimd 1889 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤))
197, 18nfald 2316 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤))
20 nfcvd 2892 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑦𝑤)
21 nfcvf2 2922 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
2221adantr 479 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑦𝑥)
2320, 22nfeqd 2902 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑦 𝑤 = 𝑥)
247, 23nfan1 2188 . . . . . . 7 𝑦((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
25 elequ2 2113 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
26 elequ1 2105 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
2725, 26anbi12d 630 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((𝑦𝑤𝑤𝑧) ↔ (𝑦𝑥𝑥𝑧)))
2827a1i 11 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑦𝑤𝑤𝑧) ↔ (𝑦𝑥𝑥𝑧))))
294, 16, 28cbvexd 2401 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤(𝑦𝑤𝑤𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
3029adantr 479 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑤(𝑦𝑤𝑤𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
3125adantl 480 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑦𝑤𝑦𝑥))
3230, 31imbi12d 343 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
3324, 32albid 2210 . . . . . 6 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
3433ex 411 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → (∀𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))))
354, 19, 34cbvexd 2401 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
361, 35mpbii 232 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
3736ex 411 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
38 nfae 2426 . . . 4 𝑦𝑥 𝑥 = 𝑦
39 nfae 2426 . . . . . 6 𝑥𝑥 𝑥 = 𝑦
40 elirrv 9621 . . . . . . . . 9 ¬ 𝑦𝑦
41 elequ2 2113 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
4240, 41mtbiri 326 . . . . . . . 8 (𝑥 = 𝑦 → ¬ 𝑦𝑥)
4342intnanrd 488 . . . . . . 7 (𝑥 = 𝑦 → ¬ (𝑦𝑥𝑥𝑧))
4443sps 2173 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ¬ (𝑦𝑥𝑥𝑧))
4539, 44nexd 2209 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
4645pm2.21d 121 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
4738, 46alrimi 2201 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
484719.8ad 2170 . 2 (∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
49 nfae 2426 . . . 4 𝑦𝑥 𝑥 = 𝑧
50 nfae 2426 . . . . . 6 𝑥𝑥 𝑥 = 𝑧
51 elirrv 9621 . . . . . . . . 9 ¬ 𝑧𝑧
52 elequ1 2105 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝑧𝑧𝑧))
5351, 52mtbiri 326 . . . . . . . 8 (𝑥 = 𝑧 → ¬ 𝑥𝑧)
5453intnand 487 . . . . . . 7 (𝑥 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
5554sps 2173 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
5650, 55nexd 2209 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
5756pm2.21d 121 . . . 4 (∀𝑥 𝑥 = 𝑧 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
5849, 57alrimi 2201 . . 3 (∀𝑥 𝑥 = 𝑧 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
595819.8ad 2170 . 2 (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
6037, 48, 59pm2.61ii 183 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1531  wex 1773  wnfc 2875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2365  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741  ax-reg 9617
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-eprel 5582  df-fr 5633
This theorem is referenced by:  zfcndun  10640  axunprim  35428
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