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Theorem iuncom 4463
 Description: Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
iuncom 𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 𝑥𝐴 𝐶
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem iuncom
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 rexcom 3080 . . . 4 (∃𝑥𝐴𝑦𝐵 𝑧𝐶 ↔ ∃𝑦𝐵𝑥𝐴 𝑧𝐶)
2 eliun 4460 . . . . 5 (𝑧 𝑦𝐵 𝐶 ↔ ∃𝑦𝐵 𝑧𝐶)
32rexbii 3023 . . . 4 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
4 eliun 4460 . . . . 5 (𝑧 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑧𝐶)
54rexbii 3023 . . . 4 (∃𝑦𝐵 𝑧 𝑥𝐴 𝐶 ↔ ∃𝑦𝐵𝑥𝐴 𝑧𝐶)
61, 3, 53bitr4i 291 . . 3 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑦𝐵 𝑧 𝑥𝐴 𝐶)
7 eliun 4460 . . 3 (𝑧 𝑥𝐴 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴 𝑧 𝑦𝐵 𝐶)
8 eliun 4460 . . 3 (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∃𝑦𝐵 𝑧 𝑥𝐴 𝐶)
96, 7, 83bitr4i 291 . 2 (𝑧 𝑥𝐴 𝑦𝐵 𝐶𝑧 𝑦𝐵 𝑥𝐴 𝐶)
109eqriv 2607 1 𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 𝑥𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ∪ ciun 4455 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-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 This theorem is referenced by: (None)
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