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Theorem iunxiun 4985
 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 4888 . . . . . . . 8 (𝑥 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑥𝐵)
21anbi1i 626 . . . . . . 7 ((𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ (∃𝑦𝐴 𝑥𝐵𝑧𝐶))
3 r19.41v 3300 . . . . . . 7 (∃𝑦𝐴 (𝑥𝐵𝑧𝐶) ↔ (∃𝑦𝐴 𝑥𝐵𝑧𝐶))
42, 3bitr4i 281 . . . . . 6 ((𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑦𝐴 (𝑥𝐵𝑧𝐶))
54exbii 1849 . . . . 5 (∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑥𝑦𝐴 (𝑥𝐵𝑧𝐶))
6 rexcom4 3212 . . . . 5 (∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶) ↔ ∃𝑥𝑦𝐴 (𝑥𝐵𝑧𝐶))
75, 6bitr4i 281 . . . 4 (∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶))
8 df-rex 3112 . . . 4 (∃𝑥 𝑦𝐴 𝐵𝑧𝐶 ↔ ∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶))
9 eliun 4888 . . . . . 6 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑧𝐶)
10 df-rex 3112 . . . . . 6 (∃𝑥𝐵 𝑧𝐶 ↔ ∃𝑥(𝑥𝐵𝑧𝐶))
119, 10bitri 278 . . . . 5 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥(𝑥𝐵𝑧𝐶))
1211rexbii 3210 . . . 4 (∃𝑦𝐴 𝑧 𝑥𝐵 𝐶 ↔ ∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶))
137, 8, 123bitr4i 306 . . 3 (∃𝑥 𝑦𝐴 𝐵𝑧𝐶 ↔ ∃𝑦𝐴 𝑧 𝑥𝐵 𝐶)
14 eliun 4888 . . 3 (𝑧 𝑥 𝑦𝐴 𝐵𝐶 ↔ ∃𝑥 𝑦𝐴 𝐵𝑧𝐶)
15 eliun 4888 . . 3 (𝑧 𝑦𝐴 𝑥𝐵 𝐶 ↔ ∃𝑦𝐴 𝑧 𝑥𝐵 𝐶)
1613, 14, 153bitr4i 306 . 2 (𝑧 𝑥 𝑦𝐴 𝐵𝐶𝑧 𝑦𝐴 𝑥𝐵 𝐶)
1716eqriv 2795 1 𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  ∪ ciun 4884 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-iun 4886 This theorem is referenced by: (None)
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