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Theorem zfcndun 9774
Description: Axiom of Union ax-un 7228, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification 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 9755 . 2 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
2 elequ2 2121 . . . . . . 7 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
3 elequ1 2114 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
42, 3anbi12d 624 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑤𝑤𝑥) ↔ (𝑧𝑦𝑦𝑥)))
54cbvexvw 2087 . . . . 5 (∃𝑤(𝑧𝑤𝑤𝑥) ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
65imbi1i 341 . . . 4 ((∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
76albii 1863 . . 3 (∀𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
87exbii 1892 . 2 (∃𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
91, 8mpbir 223 1 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wal 1599   = wceq 1601  wex 1823  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228  ax-reg 8788
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-eprel 5268  df-fr 5316
This theorem is referenced by: (None)
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