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Theorem zfcndpow 9476
 Description: Axiom of Power Sets ax-pow 4873, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4887. (Contributed by NM, 15-Aug-2003.) (Proof modification 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 4887 . . . . 5 ¬ ∀𝑦 𝑦 = 𝑧
2 exnal 1794 . . . . 5 (∃𝑦 ¬ 𝑦 = 𝑧 ↔ ¬ ∀𝑦 𝑦 = 𝑧)
31, 2mpbir 221 . . . 4 𝑦 ¬ 𝑦 = 𝑧
4 nfe1 2067 . . . . 5 𝑦𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦)
5 axpownd 9461 . . . . 5 𝑦 = 𝑧 → ∃𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦))
64, 5exlimi 2124 . . . 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 1953 . . . . . . . 8 (∃𝑥 𝑦𝑧𝑦𝑧)
9 19.3v 1954 . . . . . . . 8 (∀𝑧 𝑦𝑥𝑦𝑥)
108, 9imbi12i 339 . . . . . . 7 ((∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) ↔ (𝑦𝑧𝑦𝑥))
1110albii 1787 . . . . . 6 (∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
1211imbi1i 338 . . . . 5 ((∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ (∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
1312albii 1787 . . . 4 (∀𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ ∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
1413exbii 1814 . . 3 (∃𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
157, 14mpbi 220 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
16 elequ1 2037 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
17 elequ1 2037 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
1816, 17imbi12d 333 . . . . . 6 (𝑤 = 𝑦 → ((𝑤𝑧𝑤𝑥) ↔ (𝑦𝑧𝑦𝑥)))
1918cbvalv 2309 . . . . 5 (∀𝑤(𝑤𝑧𝑤𝑥) ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
2019imbi1i 338 . . . 4 ((∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ (∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2120albii 1787 . . 3 (∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ ∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2221exbii 1814 . 2 (∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2315, 22mpbir 221 1 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-reg 8538 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213 This theorem is referenced by: (None)
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