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Theorem iuncom4 5000
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 3071 . . . . . . 7 (∃𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(𝑧𝐵𝑦𝑧))
21rexbii 3094 . . . . . 6 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧))
3 rexcom4 3288 . . . . . 6 (∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧) ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
42, 3bitri 275 . . . . 5 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
5 r19.41v 3189 . . . . . 6 (∃𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
65exbii 1848 . . . . 5 (∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
74, 6bitri 275 . . . 4 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
8 eluni2 4911 . . . . 5 (𝑦 𝐵 ↔ ∃𝑧𝐵 𝑦𝑧)
98rexbii 3094 . . . 4 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑥𝐴𝑧𝐵 𝑦𝑧)
10 df-rex 3071 . . . . 5 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧))
11 eliun 4995 . . . . . . 7 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
1211anbi1i 624 . . . . . 6 ((𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1312exbii 1848 . . . . 5 (∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1410, 13bitri 275 . . . 4 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
157, 9, 143bitr4i 303 . . 3 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
16 eliun 4995 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 𝐵)
17 eluni2 4911 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
1815, 16, 173bitr4i 303 . 2 (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1918eqriv 2734 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070   cuni 4907   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-uni 4908  df-iun 4993
This theorem is referenced by:  ituniiun  10462  tgidm  22987  txcmplem2  23650
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