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Theorem iuncom 4890
 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 3273 . . . 4 (∃𝑥𝐴𝑦𝐵 𝑧𝐶 ↔ ∃𝑦𝐵𝑥𝐴 𝑧𝐶)
2 eliun 4887 . . . . 5 (𝑧 𝑦𝐵 𝐶 ↔ ∃𝑦𝐵 𝑧𝐶)
32rexbii 3175 . . . 4 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
4 eliun 4887 . . . . 5 (𝑧 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑧𝐶)
54rexbii 3175 . . . 4 (∃𝑦𝐵 𝑧 𝑥𝐴 𝐶 ↔ ∃𝑦𝐵𝑥𝐴 𝑧𝐶)
61, 3, 53bitr4i 306 . . 3 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑦𝐵 𝑧 𝑥𝐴 𝐶)
7 eliun 4887 . . 3 (𝑧 𝑥𝐴 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴 𝑧 𝑦𝐵 𝐶)
8 eliun 4887 . . 3 (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∃𝑦𝐵 𝑧 𝑥𝐴 𝐶)
96, 7, 83bitr4i 306 . 2 (𝑧 𝑥𝐴 𝑦𝐵 𝐶𝑧 𝑦𝐵 𝑥𝐴 𝐶)
109eqriv 2755 1 𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 𝑥𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  ∃wrex 3071  ∪ ciun 4883 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-11 2158  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-iun 4885 This theorem is referenced by: (None)
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