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Theorem iuncom 3828
 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 2599 . . . 4 (∃𝑥𝐴𝑦𝐵 𝑧𝐶 ↔ ∃𝑦𝐵𝑥𝐴 𝑧𝐶)
2 eliun 3826 . . . . 5 (𝑧 𝑦𝐵 𝐶 ↔ ∃𝑦𝐵 𝑧𝐶)
32rexbii 2446 . . . 4 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
4 eliun 3826 . . . . 5 (𝑧 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑧𝐶)
54rexbii 2446 . . . 4 (∃𝑦𝐵 𝑧 𝑥𝐴 𝐶 ↔ ∃𝑦𝐵𝑥𝐴 𝑧𝐶)
61, 3, 53bitr4i 211 . . 3 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑦𝐵 𝑧 𝑥𝐴 𝐶)
7 eliun 3826 . . 3 (𝑧 𝑥𝐴 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴 𝑧 𝑦𝐵 𝐶)
8 eliun 3826 . . 3 (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∃𝑦𝐵 𝑧 𝑥𝐴 𝐶)
96, 7, 83bitr4i 211 . 2 (𝑧 𝑥𝐴 𝑦𝐵 𝐶𝑧 𝑦𝐵 𝑥𝐴 𝐶)
109eqriv 2137 1 𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 𝑥𝐴 𝐶
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∈ wcel 1481  ∃wrex 2418  ∪ ciun 3822 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-iun 3824 This theorem is referenced by: (None)
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