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Theorem zfcndun 10656
Description: Axiom of Union ax-un 7756, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2376. (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 10637 . 2 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
2 elequ2 2122 . . . . . . 7 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
3 elequ1 2114 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
42, 3anbi12d 632 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑤𝑤𝑥) ↔ (𝑧𝑦𝑦𝑥)))
54cbvexvw 2035 . . . . 5 (∃𝑤(𝑧𝑤𝑤𝑥) ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
65imbi1i 349 . . . 4 ((∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
76albii 1818 . . 3 (∀𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
87exbii 1847 . 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 1537   = wceq 1539  wex 1778  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-reg 9633
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-eprel 5583  df-fr 5636
This theorem is referenced by: (None)
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