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Theorem axpowndlem4 9407
Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
axpowndlem4 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))

Proof of Theorem axpowndlem4
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 axpowndlem3 9406 . . . . 5 𝑥 = 𝑤 → ∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
21ax-gen 1720 . . . 4 𝑤𝑥 = 𝑤 → ∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
3 nfnae 2316 . . . . . 6 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 nfnae 2316 . . . . . 6 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
53, 4nfan 1826 . . . . 5 𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
6 nfcvf 2785 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
76adantr 481 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑥)
8 nfcvd 2763 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑤)
97, 8nfeqd 2769 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥 = 𝑤)
109nfnd 1783 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 ¬ 𝑥 = 𝑤)
11 nfnae 2316 . . . . . . . 8 𝑥 ¬ ∀𝑦 𝑦 = 𝑥
12 nfnae 2316 . . . . . . . 8 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
1311, 12nfan 1826 . . . . . . 7 𝑥(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
14 nfv 1841 . . . . . . . 8 𝑤(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
15 nfnae 2316 . . . . . . . . . . . . 13 𝑧 ¬ ∀𝑦 𝑦 = 𝑥
16 nfnae 2316 . . . . . . . . . . . . 13 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
1715, 16nfan 1826 . . . . . . . . . . . 12 𝑧(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
187, 8nfeld 2770 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥𝑤)
1917, 18nfexd 2165 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑧 𝑥𝑤)
20 nfcvf 2785 . . . . . . . . . . . . . 14 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
2120adantl 482 . . . . . . . . . . . . 13 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑧)
227, 21nfeld 2770 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥𝑧)
2314, 22nfald 2163 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤 𝑥𝑧)
2419, 23nfimd 1821 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧))
2513, 24nfald 2163 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧))
268, 7nfeld 2770 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤𝑥)
2725, 26nfimd 1821 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
2814, 27nfald 2163 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
2913, 28nfexd 2165 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
3010, 29nfimd 1821 . . . . 5 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥 = 𝑤 → ∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥)))
31 equequ2 1951 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
3231notbid 308 . . . . . . . 8 (𝑤 = 𝑦 → (¬ 𝑥 = 𝑤 ↔ ¬ 𝑥 = 𝑦))
3332adantl 482 . . . . . . 7 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (¬ 𝑥 = 𝑤 ↔ ¬ 𝑥 = 𝑦))
34 nfcvd 2763 . . . . . . . . . . . . . . 15 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑤)
35 nfcvf2 2786 . . . . . . . . . . . . . . . 16 (¬ ∀𝑦 𝑦 = 𝑥𝑥𝑦)
3635adantr 481 . . . . . . . . . . . . . . 15 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑦)
3734, 36nfeqd 2769 . . . . . . . . . . . . . 14 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥 𝑤 = 𝑦)
3813, 37nfan1 2066 . . . . . . . . . . . . 13 𝑥((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
39 nfcvd 2763 . . . . . . . . . . . . . . . . 17 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑤)
40 nfcvf2 2786 . . . . . . . . . . . . . . . . . 18 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
4140adantl 482 . . . . . . . . . . . . . . . . 17 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑦)
4239, 41nfeqd 2769 . . . . . . . . . . . . . . . 16 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑦)
4317, 42nfan1 2066 . . . . . . . . . . . . . . 15 𝑧((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
44 elequ2 2002 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
4544adantl 482 . . . . . . . . . . . . . . 15 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑥𝑤𝑥𝑦))
4643, 45exbid 2089 . . . . . . . . . . . . . 14 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑧 𝑥𝑤 ↔ ∃𝑧 𝑥𝑦))
47 biidd 252 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑦 → (𝑥𝑧𝑥𝑧))
4847a1i 11 . . . . . . . . . . . . . . . 16 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → (𝑥𝑧𝑥𝑧)))
495, 22, 48cbvald 2275 . . . . . . . . . . . . . . 15 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤 𝑥𝑧 ↔ ∀𝑦 𝑥𝑧))
5049adantr 481 . . . . . . . . . . . . . 14 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑤 𝑥𝑧 ↔ ∀𝑦 𝑥𝑧))
5146, 50imbi12d 334 . . . . . . . . . . . . 13 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
5238, 51albid 2088 . . . . . . . . . . . 12 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
53 elequ1 1995 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
5453adantl 482 . . . . . . . . . . . 12 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤𝑥𝑦𝑥))
5552, 54imbi12d 334 . . . . . . . . . . 11 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥) ↔ (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5655ex 450 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥) ↔ (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
575, 27, 56cbvald 2275 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥) ↔ ∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5813, 57exbid 2089 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥) ↔ ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5958adantr 481 . . . . . . 7 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥) ↔ ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
6033, 59imbi12d 334 . . . . . 6 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((¬ 𝑥 = 𝑤 → ∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥)) ↔ (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
6160ex 450 . . . . 5 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((¬ 𝑥 = 𝑤 → ∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥)) ↔ (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))))
625, 30, 61cbvald 2275 . . . 4 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤𝑥 = 𝑤 → ∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥)) ↔ ∀𝑦𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
632, 62mpbii 223 . . 3 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∀𝑦𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
646319.21bi 2057 . 2 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
6564ex 450 1 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479  wex 1702  wnfc 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-reg 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-pw 4151  df-sn 4169  df-pr 4171
This theorem is referenced by:  axpownd  9408
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