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Theorem axpowndlem4 10021
 Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.)
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 10020 . . . . 5 𝑥 = 𝑤 → ∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
21ax-gen 1792 . . . 4 𝑤𝑥 = 𝑤 → ∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
3 nfnae 2452 . . . . . 6 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 nfnae 2452 . . . . . 6 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
53, 4nfan 1896 . . . . 5 𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
6 nfcvf 3007 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
76adantr 483 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑥)
8 nfcvd 2978 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑤)
97, 8nfeqd 2988 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥 = 𝑤)
109nfnd 1854 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 ¬ 𝑥 = 𝑤)
11 nfnae 2452 . . . . . . . 8 𝑥 ¬ ∀𝑦 𝑦 = 𝑥
12 nfnae 2452 . . . . . . . 8 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
1311, 12nfan 1896 . . . . . . 7 𝑥(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
14 nfv 1911 . . . . . . . 8 𝑤(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
15 nfnae 2452 . . . . . . . . . . . . 13 𝑧 ¬ ∀𝑦 𝑦 = 𝑥
16 nfnae 2452 . . . . . . . . . . . . 13 𝑧 ¬ ∀𝑦 𝑦 = 𝑧
1715, 16nfan 1896 . . . . . . . . . . . 12 𝑧(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
187, 8nfeld 2989 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥𝑤)
1917, 18nfexd 2344 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑧 𝑥𝑤)
20 nfcvf 3007 . . . . . . . . . . . . . 14 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
2120adantl 484 . . . . . . . . . . . . 13 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑦𝑧)
227, 21nfeld 2989 . . . . . . . . . . . 12 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥𝑧)
2314, 22nfald 2343 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤 𝑥𝑧)
2419, 23nfimd 1891 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧))
2513, 24nfald 2343 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧))
268, 7nfeld 2989 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤𝑥)
2725, 26nfimd 1891 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
2814, 27nfald 2343 . . . . . . 7 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
2913, 28nfexd 2344 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥))
3010, 29nfimd 1891 . . . . 5 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥 = 𝑤 → ∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥)))
31 equequ2 2029 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
3231notbid 320 . . . . . . . 8 (𝑤 = 𝑦 → (¬ 𝑥 = 𝑤 ↔ ¬ 𝑥 = 𝑦))
3332adantl 484 . . . . . . 7 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (¬ 𝑥 = 𝑤 ↔ ¬ 𝑥 = 𝑦))
34 nfcvd 2978 . . . . . . . . . . . . . . 15 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑤)
35 nfcvf2 3008 . . . . . . . . . . . . . . . 16 (¬ ∀𝑦 𝑦 = 𝑥𝑥𝑦)
3635adantr 483 . . . . . . . . . . . . . . 15 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑥𝑦)
3734, 36nfeqd 2988 . . . . . . . . . . . . . 14 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥 𝑤 = 𝑦)
3813, 37nfan1 2196 . . . . . . . . . . . . 13 𝑥((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
39 nfcvd 2978 . . . . . . . . . . . . . . . . 17 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑤)
40 nfcvf2 3008 . . . . . . . . . . . . . . . . . 18 (¬ ∀𝑦 𝑦 = 𝑧𝑧𝑦)
4140adantl 484 . . . . . . . . . . . . . . . . 17 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → 𝑧𝑦)
4239, 41nfeqd 2988 . . . . . . . . . . . . . . . 16 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑦)
4317, 42nfan1 2196 . . . . . . . . . . . . . . 15 𝑧((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
44 elequ2 2125 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
4544adantl 484 . . . . . . . . . . . . . . 15 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑥𝑤𝑥𝑦))
4643, 45exbid 2221 . . . . . . . . . . . . . 14 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑧 𝑥𝑤 ↔ ∃𝑧 𝑥𝑦))
47 biidd 264 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑦 → (𝑥𝑧𝑥𝑧))
4847a1i 11 . . . . . . . . . . . . . . . 16 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → (𝑥𝑧𝑥𝑧)))
495, 22, 48cbvald 2424 . . . . . . . . . . . . . . 15 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤 𝑥𝑧 ↔ ∀𝑦 𝑥𝑧))
5049adantr 483 . . . . . . . . . . . . . 14 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑤 𝑥𝑧 ↔ ∀𝑦 𝑥𝑧))
5146, 50imbi12d 347 . . . . . . . . . . . . 13 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
5238, 51albid 2220 . . . . . . . . . . . 12 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
53 elequ1 2117 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
5453adantl 484 . . . . . . . . . . . 12 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤𝑥𝑦𝑥))
5552, 54imbi12d 347 . . . . . . . . . . 11 (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥) ↔ (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5655ex 415 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥) ↔ (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
575, 27, 56cbvald 2424 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥) ↔ ∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5813, 57exbid 2221 . . . . . . . 8 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∃𝑥𝑤(∀𝑥(∃𝑧 𝑥𝑤 → ∀𝑤 𝑥𝑧) → 𝑤𝑥) ↔ ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5958adantr 483 . . . . . . 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 347 . . . . . 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 415 . . . . 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 2424 . . . 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 235 . . 3 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∀𝑦𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
646319.21bi 2184 . 2 ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
6564ex 415 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 208   ∧ wa 398  ∀wal 1531  ∃wex 1776  Ⅎwnfc 2961 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-reg 9055 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4567  df-pr 4569 This theorem is referenced by:  axpownd  10022
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