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Theorem axpownd 10030
 Description: A version of the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 4-Jan-2002.) (New usage is discouraged.)
Assertion
Ref Expression
axpownd 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))

Proof of Theorem axpownd
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 axpowndlem4 10029 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
2 axpowndlem1 10026 . . 3 (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
32aecoms 2439 . 2 (∀𝑦 𝑦 = 𝑥 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
42a1d 25 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
5 nfnae 2445 . . . . . . . 8 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
6 nfae 2444 . . . . . . . 8 𝑦𝑦 𝑦 = 𝑧
75, 6nfan 1900 . . . . . . 7 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧)
8 el 5239 . . . . . . . . . . . . 13 𝑤 𝑥𝑤
9 nfcvf2 2982 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
10 nfcvd 2956 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑤)
119, 10nfeld 2966 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥𝑤)
12 elequ2 2126 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
1312a1i 11 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦)))
145, 11, 13cbvexd 2418 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑤 𝑥𝑤 ↔ ∃𝑦 𝑥𝑦))
158, 14mpbii 236 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦 𝑥𝑦)
161519.8ad 2179 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦 𝑥𝑦)
17 df-ex 1782 . . . . . . . . . . 11 (∃𝑥𝑦 𝑥𝑦 ↔ ¬ ∀𝑥 ¬ ∃𝑦 𝑥𝑦)
1816, 17sylib 221 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∃𝑦 𝑥𝑦)
1918adantr 484 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥 ¬ ∃𝑦 𝑥𝑦)
20 biidd 265 . . . . . . . . . . . . . 14 (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥𝑦 ↔ ¬ 𝑥𝑦))
2120dral1 2450 . . . . . . . . . . . . 13 (∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ 𝑥𝑦 ↔ ∀𝑧 ¬ 𝑥𝑦))
22 alnex 1783 . . . . . . . . . . . . 13 (∀𝑦 ¬ 𝑥𝑦 ↔ ¬ ∃𝑦 𝑥𝑦)
23 alnex 1783 . . . . . . . . . . . . 13 (∀𝑧 ¬ 𝑥𝑦 ↔ ¬ ∃𝑧 𝑥𝑦)
2421, 22, 233bitr3g 316 . . . . . . . . . . . 12 (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥𝑦 ↔ ¬ ∃𝑧 𝑥𝑦))
25 nd2 10017 . . . . . . . . . . . . 13 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑥𝑧)
26 mtt 368 . . . . . . . . . . . . 13 (¬ ∀𝑦 𝑥𝑧 → (¬ ∃𝑧 𝑥𝑦 ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
2725, 26syl 17 . . . . . . . . . . . 12 (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑧 𝑥𝑦 ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
2824, 27bitrd 282 . . . . . . . . . . 11 (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥𝑦 ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
2928dral2 2449 . . . . . . . . . 10 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 ¬ ∃𝑦 𝑥𝑦 ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
3029adantl 485 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 ¬ ∃𝑦 𝑥𝑦 ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
3119, 30mtbid 327 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧))
3231pm2.21d 121 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
337, 32alrimi 2211 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
343319.8ad 2179 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
3534a1d 25 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
3635ex 416 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
374, 36pm2.61i 185 . 2 (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
381, 3, 37pm2.61ii 186 1 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-reg 9058 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531 This theorem is referenced by:  zfcndpow  10045  axpowprim  33109
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