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Theorem zfcndun 10369
Description: Axiom of Union ax-un 7588, reproved from conditionless ZFC axioms. 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
zfcndun 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfcndun
StepHypRef Expression
1 axunnd 10350 . 2 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
2 elequ2 2121 . . . . . . 7 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
3 elequ1 2113 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
42, 3anbi12d 631 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑤𝑤𝑥) ↔ (𝑧𝑦𝑦𝑥)))
54cbvexvw 2040 . . . . 5 (∃𝑤(𝑧𝑤𝑤𝑥) ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
65imbi1i 350 . . . 4 ((∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
76albii 1822 . . 3 (∀𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
87exbii 1850 . 2 (∃𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
91, 8mpbir 230 1 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  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 5225  ax-nul 5232  ax-pr 5354  ax-un 7588  ax-reg 9349
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  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-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-br 5077  df-opab 5139  df-eprel 5497  df-fr 5546
This theorem is referenced by: (None)
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