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Theorem zfcndpow 10038
Description: Axiom of Power Sets ax-pow 5254, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness" dtru 5259. Usage of this theorem is discouraged because it depends on ax-13 2392. (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 5259 . . . . 5 ¬ ∀𝑦 𝑦 = 𝑧
2 exnal 1828 . . . . 5 (∃𝑦 ¬ 𝑦 = 𝑧 ↔ ¬ ∀𝑦 𝑦 = 𝑧)
31, 2mpbir 234 . . . 4 𝑦 ¬ 𝑦 = 𝑧
4 nfe1 2155 . . . . 5 𝑦𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦)
5 axpownd 10023 . . . . 5 𝑦 = 𝑧 → ∃𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦))
64, 5exlimi 2219 . . . 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 1989 . . . . . . . 8 (∃𝑥 𝑦𝑧𝑦𝑧)
9 19.3v 1987 . . . . . . . 8 (∀𝑧 𝑦𝑥𝑦𝑥)
108, 9imbi12i 354 . . . . . . 7 ((∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) ↔ (𝑦𝑧𝑦𝑥))
1110albii 1821 . . . . . 6 (∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
1211imbi1i 353 . . . . 5 ((∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ (∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
1312albii 1821 . . . 4 (∀𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ ∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
1413exbii 1849 . . 3 (∃𝑦𝑧(∀𝑦(∃𝑥 𝑦𝑧 → ∀𝑧 𝑦𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
157, 14mpbi 233 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
16 elequ1 2122 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
17 elequ1 2122 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
1816, 17imbi12d 348 . . . . . 6 (𝑤 = 𝑦 → ((𝑤𝑧𝑤𝑥) ↔ (𝑦𝑧𝑦𝑥)))
1918cbvalvw 2044 . . . . 5 (∀𝑤(𝑤𝑧𝑤𝑥) ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
2019imbi1i 353 . . . 4 ((∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ (∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2120albii 1821 . . 3 (∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ ∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2221exbii 1849 . 2 (∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦))
2315, 22mpbir 234 1 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536   = wceq 1538  wex 1781  wcel 2115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-reg 9055
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-sn 4551  df-pr 4553
This theorem is referenced by: (None)
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