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Theorem zfcndun 10653
Description: Axiom of Union ax-un 7754, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2375. (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 10634 . 2 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
2 elequ2 2121 . . . . . . 7 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
3 elequ1 2113 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
42, 3anbi12d 632 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑤𝑤𝑥) ↔ (𝑧𝑦𝑦𝑥)))
54cbvexvw 2034 . . . . 5 (∃𝑤(𝑧𝑤𝑤𝑥) ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
65imbi1i 349 . . . 4 ((∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
76albii 1816 . . 3 (∀𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
87exbii 1845 . 2 (∃𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
91, 8mpbir 231 1 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-fr 5641
This theorem is referenced by: (None)
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