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Theorem eliuniin 38307
 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 . . . . . . 7 𝐴 = 𝑥𝐵 𝑦𝐶 𝐷
21eleq2i 2680 . . . . . 6 (𝑍𝐴𝑍 𝑥𝐵 𝑦𝐶 𝐷)
32biimpi 205 . . . . 5 (𝑍𝐴𝑍 𝑥𝐵 𝑦𝐶 𝐷)
4 eliun 4460 . . . . 5 (𝑍 𝑥𝐵 𝑦𝐶 𝐷 ↔ ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
53, 4sylib 207 . . . 4 (𝑍𝐴 → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
6 id 22 . . . . . . 7 (𝑍 𝑦𝐶 𝐷𝑍 𝑦𝐶 𝐷)
7 eliin 4461 . . . . . . 7 (𝑍 𝑦𝐶 𝐷 → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
86, 7mpbid 221 . . . . . 6 (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷)
98a1i 11 . . . . 5 (𝑍𝐴 → (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷))
109reximdv 2999 . . . 4 (𝑍𝐴 → (∃𝑥𝐵 𝑍 𝑦𝐶 𝐷 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
115, 10mpd 15 . . 3 (𝑍𝐴 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
1211a1i 11 . 2 (𝑍𝑉 → (𝑍𝐴 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
13 simp2 1055 . . . . . . 7 ((𝑍𝑉𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑥𝐵)
14 simpr 476 . . . . . . . . 9 ((𝑍𝑉 ∧ ∀𝑦𝐶 𝑍𝐷) → ∀𝑦𝐶 𝑍𝐷)
15 id 22 . . . . . . . . . . 11 (𝑍𝑉𝑍𝑉)
16 eliin 4461 . . . . . . . . . . 11 (𝑍𝑉 → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
1715, 16syl 17 . . . . . . . . . 10 (𝑍𝑉 → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
1817adantr 480 . . . . . . . . 9 ((𝑍𝑉 ∧ ∀𝑦𝐶 𝑍𝐷) → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
1914, 18mpbird 246 . . . . . . . 8 ((𝑍𝑉 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑦𝐶 𝐷)
20193adant2 1073 . . . . . . 7 ((𝑍𝑉𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑦𝐶 𝐷)
21 rspe 2986 . . . . . . 7 ((𝑥𝐵𝑍 𝑦𝐶 𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
2213, 20, 21syl2anc 691 . . . . . 6 ((𝑍𝑉𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
2322, 4sylibr 223 . . . . 5 ((𝑍𝑉𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑥𝐵 𝑦𝐶 𝐷)
2423, 2sylibr 223 . . . 4 ((𝑍𝑉𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍𝐴)
25243exp 1256 . . 3 (𝑍𝑉 → (𝑥𝐵 → (∀𝑦𝐶 𝑍𝐷𝑍𝐴)))
2625rexlimdv 3012 . 2 (𝑍𝑉 → (∃𝑥𝐵𝑦𝐶 𝑍𝐷𝑍𝐴))
2712, 26impbid 201 1 (𝑍𝑉 → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∪ ciun 4455  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-iun 4457  df-iin 4458 This theorem is referenced by: (None)
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