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Theorem axpownd 9680
Description: A version of the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
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 9679 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
2 axpowndlem1 9676 . . 3 (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
32aecoms 2408 . 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 2414 . . . . . . . 8 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
6 nfae 2412 . . . . . . . 8 𝑦𝑦 𝑦 = 𝑧
75, 6nfan 1998 . . . . . . 7 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧)
8 el 5007 . . . . . . . . . . . . 13 𝑤 𝑥𝑤
9 nfcvf2 2932 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
10 nfcvd 2908 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑤)
119, 10nfeld 2916 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥𝑤)
12 elequ2 2169 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
1312a1i 11 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦)))
145, 11, 13cbvexd 2381 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑤 𝑥𝑤 ↔ ∃𝑦 𝑥𝑦))
158, 14mpbii 224 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦 𝑥𝑦)
16 19.8a 2214 . . . . . . . . . . . 12 (∃𝑦 𝑥𝑦 → ∃𝑥𝑦 𝑥𝑦)
1715, 16syl 17 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦 𝑥𝑦)
18 df-ex 1875 . . . . . . . . . . 11 (∃𝑥𝑦 𝑥𝑦 ↔ ¬ ∀𝑥 ¬ ∃𝑦 𝑥𝑦)
1917, 18sylib 209 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∃𝑦 𝑥𝑦)
2019adantr 472 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥 ¬ ∃𝑦 𝑥𝑦)
21 biidd 253 . . . . . . . . . . . . . 14 (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥𝑦 ↔ ¬ 𝑥𝑦))
2221dral1 2419 . . . . . . . . . . . . 13 (∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ 𝑥𝑦 ↔ ∀𝑧 ¬ 𝑥𝑦))
23 alnex 1876 . . . . . . . . . . . . 13 (∀𝑦 ¬ 𝑥𝑦 ↔ ¬ ∃𝑦 𝑥𝑦)
24 alnex 1876 . . . . . . . . . . . . 13 (∀𝑧 ¬ 𝑥𝑦 ↔ ¬ ∃𝑧 𝑥𝑦)
2522, 23, 243bitr3g 304 . . . . . . . . . . . 12 (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥𝑦 ↔ ¬ ∃𝑧 𝑥𝑦))
26 nd2 9667 . . . . . . . . . . . . 13 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑥𝑧)
27 mtt 355 . . . . . . . . . . . . 13 (¬ ∀𝑦 𝑥𝑧 → (¬ ∃𝑧 𝑥𝑦 ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
2826, 27syl 17 . . . . . . . . . . . 12 (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑧 𝑥𝑦 ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
2925, 28bitrd 270 . . . . . . . . . . 11 (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥𝑦 ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
3029dral2 2418 . . . . . . . . . 10 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 ¬ ∃𝑦 𝑥𝑦 ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
3130adantl 473 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 ¬ ∃𝑦 𝑥𝑦 ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
3220, 31mtbid 315 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧))
3332pm2.21d 119 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
347, 33alrimi 2246 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
35 19.8a 2214 . . . . . 6 (∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
3634, 35syl 17 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
3736a1d 25 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
3837ex 401 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
394, 38pm2.61i 176 . 2 (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
401, 3, 39pm2.61ii 177 1 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1650  wex 1874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-reg 8708
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-pw 4319  df-sn 4337  df-pr 4339
This theorem is referenced by:  zfcndpow  9695  axpowprim  32048
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