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Theorem iuncom4 4890
 Description: Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
Assertion
Ref Expression
iuncom4 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵

Proof of Theorem iuncom4
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 3112 . . . . . . 7 (∃𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(𝑧𝐵𝑦𝑧))
21rexbii 3210 . . . . . 6 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧))
3 rexcom4 3212 . . . . . 6 (∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧) ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
42, 3bitri 278 . . . . 5 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
5 r19.41v 3300 . . . . . 6 (∃𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
65exbii 1849 . . . . 5 (∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
74, 6bitri 278 . . . 4 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
8 eluni2 4805 . . . . 5 (𝑦 𝐵 ↔ ∃𝑧𝐵 𝑦𝑧)
98rexbii 3210 . . . 4 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑥𝐴𝑧𝐵 𝑦𝑧)
10 df-rex 3112 . . . . 5 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧))
11 eliun 4886 . . . . . . 7 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
1211anbi1i 626 . . . . . 6 ((𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1312exbii 1849 . . . . 5 (∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1410, 13bitri 278 . . . 4 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
157, 9, 143bitr4i 306 . . 3 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
16 eliun 4886 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 𝐵)
17 eluni2 4805 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
1815, 16, 173bitr4i 306 . 2 (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1918eqriv 2795 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  ∪ cuni 4801  ∪ ciun 4882 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-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-uni 4802  df-iun 4884 This theorem is referenced by:  ituniiun  9840  tgidm  21599  txcmplem2  22261
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