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Theorem iunxiun 4752
 Description: Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
iunxiun 𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝑦,𝐶
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥)

Proof of Theorem iunxiun
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eliun 4668 . . . . . . . 8 (𝑥 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑥𝐵)
21anbi1i 733 . . . . . . 7 ((𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ (∃𝑦𝐴 𝑥𝐵𝑧𝐶))
3 r19.41v 3219 . . . . . . 7 (∃𝑦𝐴 (𝑥𝐵𝑧𝐶) ↔ (∃𝑦𝐴 𝑥𝐵𝑧𝐶))
42, 3bitr4i 267 . . . . . 6 ((𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑦𝐴 (𝑥𝐵𝑧𝐶))
54exbii 1915 . . . . 5 (∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑥𝑦𝐴 (𝑥𝐵𝑧𝐶))
6 rexcom4 3357 . . . . 5 (∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶) ↔ ∃𝑥𝑦𝐴 (𝑥𝐵𝑧𝐶))
75, 6bitr4i 267 . . . 4 (∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶))
8 df-rex 3048 . . . 4 (∃𝑥 𝑦𝐴 𝐵𝑧𝐶 ↔ ∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶))
9 eliun 4668 . . . . . 6 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑧𝐶)
10 df-rex 3048 . . . . . 6 (∃𝑥𝐵 𝑧𝐶 ↔ ∃𝑥(𝑥𝐵𝑧𝐶))
119, 10bitri 264 . . . . 5 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥(𝑥𝐵𝑧𝐶))
1211rexbii 3171 . . . 4 (∃𝑦𝐴 𝑧 𝑥𝐵 𝐶 ↔ ∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶))
137, 8, 123bitr4i 292 . . 3 (∃𝑥 𝑦𝐴 𝐵𝑧𝐶 ↔ ∃𝑦𝐴 𝑧 𝑥𝐵 𝐶)
14 eliun 4668 . . 3 (𝑧 𝑥 𝑦𝐴 𝐵𝐶 ↔ ∃𝑥 𝑦𝐴 𝐵𝑧𝐶)
15 eliun 4668 . . 3 (𝑧 𝑦𝐴 𝑥𝐵 𝐶 ↔ ∃𝑦𝐴 𝑧 𝑥𝐵 𝐶)
1613, 14, 153bitr4i 292 . 2 (𝑧 𝑥 𝑦𝐴 𝐵𝐶𝑧 𝑦𝐴 𝑥𝐵 𝐶)
1716eqriv 2749 1 𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1624  ∃wex 1845   ∈ wcel 2131  ∃wrex 3043  ∪ ciun 4664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-iun 4666 This theorem is referenced by: (None)
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