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Theorem iuncom4 3743
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 2366 . . . . . . 7 (∃𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(𝑧𝐵𝑦𝑧))
21rexbii 2386 . . . . . 6 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧))
3 rexcom4 2643 . . . . . 6 (∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧) ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
42, 3bitri 183 . . . . 5 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
5 r19.41v 2524 . . . . . 6 (∃𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
65exbii 1542 . . . . 5 (∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
74, 6bitri 183 . . . 4 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
8 eluni2 3663 . . . . 5 (𝑦 𝐵 ↔ ∃𝑧𝐵 𝑦𝑧)
98rexbii 2386 . . . 4 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑥𝐴𝑧𝐵 𝑦𝑧)
10 df-rex 2366 . . . . 5 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧))
11 eliun 3740 . . . . . . 7 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
1211anbi1i 447 . . . . . 6 ((𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1312exbii 1542 . . . . 5 (∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1410, 13bitri 183 . . . 4 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
157, 9, 143bitr4i 211 . . 3 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
16 eliun 3740 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 𝐵)
17 eluni2 3663 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
1815, 16, 173bitr4i 211 . 2 (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1918eqriv 2086 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1290  wex 1427  wcel 1439  wrex 2361   cuni 3659   ciun 3736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-uni 3660  df-iun 3738
This theorem is referenced by:  tgidm  11835
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