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Theorem zfcndpow 10382
Description: Axiom of Power Sets ax-pow 5286, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness" dtru 5357. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
zfcndpow 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfcndpow
StepHypRef Expression
1 dtru 5357 . . . . 5 ¬ ∀𝑦 𝑦 = 𝑧
2 exnal 1829 . . . . 5 (∃𝑦 ¬ 𝑦 = 𝑧 ↔ ¬ ∀𝑦 𝑦 = 𝑧)
31, 2mpbir 230 . . . 4 𝑦 ¬ 𝑦 = 𝑧
4 nfe1 2147 . . . . 5 𝑦𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦)
5 axpownd 10367 . . . . 5 𝑦 = 𝑧 → ∃𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦))
64, 5exlimi 2210 . . . 4 (∃𝑦 ¬ 𝑦 = 𝑧 → ∃𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦))
73, 6ax-mp 5 . . 3 𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦)
8 19.9v 1987 . . . . . . . 8 (∃𝑥 𝑦𝑧𝑦𝑧)
9 19.3v 1985 . . . . . . . 8 (∀𝑧 𝑦𝑥𝑦𝑥)
108, 9imbi12i 351 . . . . . . 7 ((∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) ↔ (𝑦𝑧𝑦𝑥))
1110albii 1822 . . . . . 6 (∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
1211imbi1i 350 . . . . 5 ((∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ (∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
1312albii 1822 . . . 4 (∀𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ ∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
1413exbii 1850 . . 3 (∃𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
157, 14mpbi 229 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
16 elequ1 2113 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
17 elequ1 2113 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
1816, 17imbi12d 345 . . . . . 6 (𝑤 = 𝑦 → ((𝑤𝑧𝑤𝑥) ↔ (𝑦𝑧𝑦𝑥)))
1918cbvalvw 2039 . . . . 5 (∀𝑤(𝑤𝑧𝑤𝑥) ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
2019imbi1i 350 . . . 4 ((∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ (∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2120albii 1822 . . 3 (∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ ∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2221exbii 1850 . 2 (∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2315, 22mpbir 230 1 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1537   = wceq 1539  wex 1782  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-reg 9338
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564
This theorem is referenced by: (None)
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