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Theorem iunxiun 4743
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 4659 . . . . . . . 8 (𝑥 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑥𝐵)
21anbi1i 610 . . . . . . 7 ((𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ (∃𝑦𝐴 𝑥𝐵𝑧𝐶))
3 r19.41v 3237 . . . . . . 7 (∃𝑦𝐴 (𝑥𝐵𝑧𝐶) ↔ (∃𝑦𝐴 𝑥𝐵𝑧𝐶))
42, 3bitr4i 267 . . . . . 6 ((𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑦𝐴 (𝑥𝐵𝑧𝐶))
54exbii 1924 . . . . 5 (∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑥𝑦𝐴 (𝑥𝐵𝑧𝐶))
6 rexcom4 3377 . . . . 5 (∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶) ↔ ∃𝑥𝑦𝐴 (𝑥𝐵𝑧𝐶))
75, 6bitr4i 267 . . . 4 (∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶) ↔ ∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶))
8 df-rex 3067 . . . 4 (∃𝑥 𝑦𝐴 𝐵𝑧𝐶 ↔ ∃𝑥(𝑥 𝑦𝐴 𝐵𝑧𝐶))
9 eliun 4659 . . . . . 6 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑧𝐶)
10 df-rex 3067 . . . . . 6 (∃𝑥𝐵 𝑧𝐶 ↔ ∃𝑥(𝑥𝐵𝑧𝐶))
119, 10bitri 264 . . . . 5 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥(𝑥𝐵𝑧𝐶))
1211rexbii 3189 . . . 4 (∃𝑦𝐴 𝑧 𝑥𝐵 𝐶 ↔ ∃𝑦𝐴𝑥(𝑥𝐵𝑧𝐶))
137, 8, 123bitr4i 292 . . 3 (∃𝑥 𝑦𝐴 𝐵𝑧𝐶 ↔ ∃𝑦𝐴 𝑧 𝑥𝐵 𝐶)
14 eliun 4659 . . 3 (𝑧 𝑥 𝑦𝐴 𝐵𝐶 ↔ ∃𝑥 𝑦𝐴 𝐵𝑧𝐶)
15 eliun 4659 . . 3 (𝑧 𝑦𝐴 𝑥𝐵 𝐶 ↔ ∃𝑦𝐴 𝑧 𝑥𝐵 𝐶)
1613, 14, 153bitr4i 292 . 2 (𝑧 𝑥 𝑦𝐴 𝐵𝐶𝑧 𝑦𝐴 𝑥𝐵 𝐶)
1716eqriv 2768 1 𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wex 1852  wcel 2145  wrex 3062   ciun 4655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-iun 4657
This theorem is referenced by: (None)
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