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Theorem eliuniin 43870
Description: Indexed union of indexed intersections. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
eliuniin.1 𝐴 = 𝑥𝐵 𝑦𝐶 𝐷
Assertion
Ref Expression
eliuniin (𝑍𝑉 → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝑥,𝑍   𝑦,𝑍
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem eliuniin
StepHypRef Expression
1 eliuniin.1 . . . . 5 𝐴 = 𝑥𝐵 𝑦𝐶 𝐷
21eleq2i 2825 . . . 4 (𝑍𝐴𝑍 𝑥𝐵 𝑦𝐶 𝐷)
3 eliun 5001 . . . 4 (𝑍 𝑥𝐵 𝑦𝐶 𝐷 ↔ ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
42, 3sylbb 218 . . 3 (𝑍𝐴 → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
5 eliin 5002 . . . . . 6 (𝑍 𝑦𝐶 𝐷 → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
65ibi 266 . . . . 5 (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷)
76a1i 11 . . . 4 (𝑍𝐴 → (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷))
87reximdv 3170 . . 3 (𝑍𝐴 → (∃𝑥𝐵 𝑍 𝑦𝐶 𝐷 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
94, 8mpd 15 . 2 (𝑍𝐴 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
10 simp2 1137 . . . . . 6 ((𝑍𝑉𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑥𝐵)
11 eliin 5002 . . . . . . 7 (𝑍𝑉 → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
1211biimpar 478 . . . . . 6 ((𝑍𝑉 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑦𝐶 𝐷)
13 rspe 3246 . . . . . 6 ((𝑥𝐵𝑍 𝑦𝐶 𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
1410, 12, 133imp3i2an 1345 . . . . 5 ((𝑍𝑉𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
1514, 3sylibr 233 . . . 4 ((𝑍𝑉𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑥𝐵 𝑦𝐶 𝐷)
1615, 2sylibr 233 . . 3 ((𝑍𝑉𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍𝐴)
1716rexlimdv3a 3159 . 2 (𝑍𝑉 → (∃𝑥𝐵𝑦𝐶 𝑍𝐷𝑍𝐴))
189, 17impbid2 225 1 (𝑍𝑉 → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070   ciun 4997   ciin 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-v 3476  df-iun 4999  df-iin 5000
This theorem is referenced by: (None)
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