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Theorem axpownd 9625
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 9624 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
2 axpowndlem1 9621 . . 3 (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
32aecoms 2464 . 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 2470 . . . . . . . 8 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
6 nfae 2468 . . . . . . . 8 𝑦𝑦 𝑦 = 𝑧
75, 6nfan 1980 . . . . . . 7 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧)
8 el 4978 . . . . . . . . . . . . 13 𝑤 𝑥𝑤
9 nfcvf2 2938 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
10 nfcvd 2914 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑤)
119, 10nfeld 2922 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥𝑤)
12 elequ2 2159 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
1312a1i 11 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦)))
145, 11, 13cbvexd 2437 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑤 𝑥𝑤 ↔ ∃𝑦 𝑥𝑦))
158, 14mpbii 223 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦 𝑥𝑦)
16 19.8a 2206 . . . . . . . . . . . 12 (∃𝑦 𝑥𝑦 → ∃𝑥𝑦 𝑥𝑦)
1715, 16syl 17 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦 𝑥𝑦)
18 df-ex 1853 . . . . . . . . . . 11 (∃𝑥𝑦 𝑥𝑦 ↔ ¬ ∀𝑥 ¬ ∃𝑦 𝑥𝑦)
1917, 18sylib 208 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∃𝑦 𝑥𝑦)
2019adantr 466 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥 ¬ ∃𝑦 𝑥𝑦)
21 biidd 252 . . . . . . . . . . . . . 14 (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥𝑦 ↔ ¬ 𝑥𝑦))
2221dral1 2475 . . . . . . . . . . . . 13 (∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ 𝑥𝑦 ↔ ∀𝑧 ¬ 𝑥𝑦))
23 alnex 1854 . . . . . . . . . . . . 13 (∀𝑦 ¬ 𝑥𝑦 ↔ ¬ ∃𝑦 𝑥𝑦)
24 alnex 1854 . . . . . . . . . . . . 13 (∀𝑧 ¬ 𝑥𝑦 ↔ ¬ ∃𝑧 𝑥𝑦)
2522, 23, 243bitr3g 302 . . . . . . . . . . . 12 (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥𝑦 ↔ ¬ ∃𝑧 𝑥𝑦))
26 nd2 9612 . . . . . . . . . . . . 13 (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑥𝑧)
27 mtt 353 . . . . . . . . . . . . 13 (¬ ∀𝑦 𝑥𝑧 → (¬ ∃𝑧 𝑥𝑦 ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
2826, 27syl 17 . . . . . . . . . . . 12 (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑧 𝑥𝑦 ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
2925, 28bitrd 268 . . . . . . . . . . 11 (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥𝑦 ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
3029dral2 2474 . . . . . . . . . 10 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 ¬ ∃𝑦 𝑥𝑦 ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
3130adantl 467 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 ¬ ∃𝑦 𝑥𝑦 ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
3220, 31mtbid 313 . . . . . . . 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 2238 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
35 19.8a 2206 . . . . . 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 397 . . 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 196  wa 382  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-reg 8653
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319
This theorem is referenced by:  zfcndpow  9640  axpowprim  31919
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