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Theorem eliuniin2 45059
Description: Indexed union of indexed intersections. See eliincex 45049 for a counterexample showing that the precondition 𝐶 ≠ ∅ cannot be simply dropped. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
eliuniin2.1 𝑥𝐶
eliuniin2.2 𝐴 = 𝑥𝐵 𝑦𝐶 𝐷
Assertion
Ref Expression
eliuniin2 (𝐶 ≠ ∅ → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐶   𝑥,𝑍   𝑦,𝑍
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥)   𝐷(𝑥,𝑦)

Proof of Theorem eliuniin2
StepHypRef Expression
1 eliuniin2.2 . . . . 5 𝐴 = 𝑥𝐵 𝑦𝐶 𝐷
21eleq2i 2830 . . . 4 (𝑍𝐴𝑍 𝑥𝐵 𝑦𝐶 𝐷)
3 eliun 4999 . . . 4 (𝑍 𝑥𝐵 𝑦𝐶 𝐷 ↔ ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
42, 3sylbb 219 . . 3 (𝑍𝐴 → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
5 eliin 5000 . . . . . 6 (𝑍 𝑦𝐶 𝐷 → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
65ibi 267 . . . . 5 (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷)
76a1i 11 . . . 4 (𝑍𝐴 → (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷))
87reximdv 3167 . . 3 (𝑍𝐴 → (∃𝑥𝐵 𝑍 𝑦𝐶 𝐷 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
94, 8mpd 15 . 2 (𝑍𝐴 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
10 eliuniin2.1 . . . 4 𝑥𝐶
11 nfcv 2902 . . . 4 𝑥
1210, 11nfne 3040 . . 3 𝑥 𝐶 ≠ ∅
13 nfv 1911 . . 3 𝑥 𝑍𝐴
14 simp2 1136 . . . . . . 7 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑥𝐵)
15 eliin2 45055 . . . . . . . 8 (𝐶 ≠ ∅ → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
1615biimpar 477 . . . . . . 7 ((𝐶 ≠ ∅ ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑦𝐶 𝐷)
17 rspe 3246 . . . . . . 7 ((𝑥𝐵𝑍 𝑦𝐶 𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
1814, 16, 173imp3i2an 1344 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
1918, 3sylibr 234 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑥𝐵 𝑦𝐶 𝐷)
2019, 2sylibr 234 . . . 4 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍𝐴)
21203exp 1118 . . 3 (𝐶 ≠ ∅ → (𝑥𝐵 → (∀𝑦𝐶 𝑍𝐷𝑍𝐴)))
2212, 13, 21rexlimd 3263 . 2 (𝐶 ≠ ∅ → (∃𝑥𝐵𝑦𝐶 𝑍𝐷𝑍𝐴))
239, 22impbid2 226 1 (𝐶 ≠ ∅ → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1536  wcel 2105  wnfc 2887  wne 2937  wral 3058  wrex 3067  c0 4338   ciun 4995   ciin 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-nul 4339  df-iun 4997  df-iin 4998
This theorem is referenced by: (None)
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