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Theorem eliuniin2 39129
Description: Indexed union of indexed intersections. See eliincex 39119 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 2692 . . . 4 (𝑍𝐴𝑍 𝑥𝐵 𝑦𝐶 𝐷)
3 eliun 4522 . . . 4 (𝑍 𝑥𝐵 𝑦𝐶 𝐷 ↔ ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
42, 3sylbb 209 . . 3 (𝑍𝐴 → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
5 eliin 4523 . . . . . 6 (𝑍 𝑦𝐶 𝐷 → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
65ibi 256 . . . . 5 (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷)
76a1i 11 . . . 4 (𝑍𝐴 → (𝑍 𝑦𝐶 𝐷 → ∀𝑦𝐶 𝑍𝐷))
87reximdv 3015 . . 3 (𝑍𝐴 → (∃𝑥𝐵 𝑍 𝑦𝐶 𝐷 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
94, 8mpd 15 . 2 (𝑍𝐴 → ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
10 eliuniin2.1 . . . 4 𝑥𝐶
11 nfcv 2763 . . . 4 𝑥
1210, 11nfne 2893 . . 3 𝑥 𝐶 ≠ ∅
13 nfv 1842 . . 3 𝑥 𝑍𝐴
14 simp2 1061 . . . . . . 7 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑥𝐵)
15 eliin2 39125 . . . . . . . 8 (𝐶 ≠ ∅ → (𝑍 𝑦𝐶 𝐷 ↔ ∀𝑦𝐶 𝑍𝐷))
1615biimpar 502 . . . . . . 7 ((𝐶 ≠ ∅ ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑦𝐶 𝐷)
17 rspe 3002 . . . . . . 7 ((𝑥𝐵𝑍 𝑦𝐶 𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
1814, 16, 173imp3i2an 1277 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → ∃𝑥𝐵 𝑍 𝑦𝐶 𝐷)
1918, 3sylibr 224 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍 𝑥𝐵 𝑦𝐶 𝐷)
2019, 2sylibr 224 . . . 4 ((𝐶 ≠ ∅ ∧ 𝑥𝐵 ∧ ∀𝑦𝐶 𝑍𝐷) → 𝑍𝐴)
21203exp 1263 . . 3 (𝐶 ≠ ∅ → (𝑥𝐵 → (∀𝑦𝐶 𝑍𝐷𝑍𝐴)))
2212, 13, 21rexlimd 3024 . 2 (𝐶 ≠ ∅ → (∃𝑥𝐵𝑦𝐶 𝑍𝐷𝑍𝐴))
239, 22impbid2 216 1 (𝐶 ≠ ∅ → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1037   = wceq 1482  wcel 1989  wnfc 2750  wne 2793  wral 2911  wrex 2912  c0 3913   ciun 4518   ciin 4519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-nul 3914  df-iun 4520  df-iin 4521
This theorem is referenced by: (None)
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