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Theorem zfcndpow 9294
Description: Axiom of Power Sets ax-pow 4764, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4778. (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 4778 . . . . 5 ¬ ∀𝑦 𝑦 = 𝑧
2 exnal 1743 . . . . 5 (∃𝑦 ¬ 𝑦 = 𝑧 ↔ ¬ ∀𝑦 𝑦 = 𝑧)
31, 2mpbir 219 . . . 4 𝑦 ¬ 𝑦 = 𝑧
4 nfe1 2013 . . . . 5 𝑦𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦)
5 axpownd 9279 . . . . 5 𝑦 = 𝑧 → ∃𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦))
64, 5exlimi 2072 . . . 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 1882 . . . . . . . 8 (∃𝑥 𝑦𝑧𝑦𝑧)
9 19.3v 1883 . . . . . . . 8 (∀𝑧 𝑦𝑥𝑦𝑥)
108, 9imbi12i 338 . . . . . . 7 ((∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) ↔ (𝑦𝑧𝑦𝑥))
1110albii 1736 . . . . . 6 (∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
1211imbi1i 337 . . . . 5 ((∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ (∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
1312albii 1736 . . . 4 (∀𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ ∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
1413exbii 1763 . . 3 (∃𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
157, 14mpbi 218 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
16 elequ1 1983 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
17 elequ1 1983 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
1816, 17imbi12d 332 . . . . . 6 (𝑤 = 𝑦 → ((𝑤𝑧𝑤𝑥) ↔ (𝑦𝑧𝑦𝑥)))
1918cbvalv 2260 . . . . 5 (∀𝑤(𝑤𝑧𝑤𝑥) ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
2019imbi1i 337 . . . 4 ((∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ (∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2120albii 1736 . . 3 (∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ ∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2221exbii 1763 . 2 (∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2315, 22mpbir 219 1 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472   = wceq 1474  wex 1694  wcel 1976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-reg 8357
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-pw 4109  df-sn 4125  df-pr 4127
This theorem is referenced by: (None)
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