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Theorem iuncom4 3691
 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 2329 . . . . . . 7 (∃𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(𝑧𝐵𝑦𝑧))
21rexbii 2348 . . . . . 6 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧))
3 rexcom4 2594 . . . . . 6 (∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧) ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
42, 3bitri 177 . . . . 5 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
5 r19.41v 2483 . . . . . 6 (∃𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
65exbii 1512 . . . . 5 (∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
74, 6bitri 177 . . . 4 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
8 eluni2 3611 . . . . 5 (𝑦 𝐵 ↔ ∃𝑧𝐵 𝑦𝑧)
98rexbii 2348 . . . 4 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑥𝐴𝑧𝐵 𝑦𝑧)
10 df-rex 2329 . . . . 5 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧))
11 eliun 3688 . . . . . . 7 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
1211anbi1i 439 . . . . . 6 ((𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1312exbii 1512 . . . . 5 (∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1410, 13bitri 177 . . . 4 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
157, 9, 143bitr4i 205 . . 3 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
16 eliun 3688 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 𝐵)
17 eluni2 3611 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
1815, 16, 173bitr4i 205 . 2 (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1918eqriv 2053 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   = wceq 1259  ∃wex 1397   ∈ wcel 1409  ∃wrex 2324  ∪ cuni 3607  ∪ ciun 3684 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-uni 3608  df-iun 3686 This theorem is referenced by: (None)
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