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Theorem iuncom 5003
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 3287 . . . 4 (∃𝑥𝐴𝑦𝐵 𝑧𝐶 ↔ ∃𝑦𝐵𝑥𝐴 𝑧𝐶)
2 eliun 4999 . . . . 5 (𝑧 𝑦𝐵 𝐶 ↔ ∃𝑦𝐵 𝑧𝐶)
32rexbii 3091 . . . 4 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
4 eliun 4999 . . . . 5 (𝑧 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑧𝐶)
54rexbii 3091 . . . 4 (∃𝑦𝐵 𝑧 𝑥𝐴 𝐶 ↔ ∃𝑦𝐵𝑥𝐴 𝑧𝐶)
61, 3, 53bitr4i 303 . . 3 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑦𝐵 𝑧 𝑥𝐴 𝐶)
7 eliun 4999 . . 3 (𝑧 𝑥𝐴 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴 𝑧 𝑦𝐵 𝐶)
8 eliun 4999 . . 3 (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∃𝑦𝐵 𝑧 𝑥𝐴 𝐶)
96, 7, 83bitr4i 303 . 2 (𝑧 𝑥𝐴 𝑦𝐵 𝐶𝑧 𝑦𝐵 𝑥𝐴 𝐶)
109eqriv 2731 1 𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 𝑥𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  wrex 3067   ciun 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-iun 4997
This theorem is referenced by:  pzriprnglem11  21519
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