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Theorem iuncom4 4499
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 2913 . . . . . . 7 (∃𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(𝑧𝐵𝑦𝑧))
21rexbii 3035 . . . . . 6 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧))
3 rexcom4 3214 . . . . . 6 (∃𝑥𝐴𝑧(𝑧𝐵𝑦𝑧) ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
42, 3bitri 264 . . . . 5 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧))
5 r19.41v 3082 . . . . . 6 (∃𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
65exbii 1771 . . . . 5 (∃𝑧𝑥𝐴 (𝑧𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
74, 6bitri 264 . . . 4 (∃𝑥𝐴𝑧𝐵 𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
8 eluni2 4411 . . . . 5 (𝑦 𝐵 ↔ ∃𝑧𝐵 𝑦𝑧)
98rexbii 3035 . . . 4 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑥𝐴𝑧𝐵 𝑦𝑧)
10 df-rex 2913 . . . . 5 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧))
11 eliun 4495 . . . . . . 7 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
1211anbi1i 730 . . . . . 6 ((𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ (∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1312exbii 1771 . . . . 5 (∃𝑧(𝑧 𝑥𝐴 𝐵𝑦𝑧) ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
1410, 13bitri 264 . . . 4 (∃𝑧 𝑥𝐴 𝐵𝑦𝑧 ↔ ∃𝑧(∃𝑥𝐴 𝑧𝐵𝑦𝑧))
157, 9, 143bitr4i 292 . . 3 (∃𝑥𝐴 𝑦 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
16 eliun 4495 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 𝐵)
17 eluni2 4411 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑧 𝑥𝐴 𝐵𝑦𝑧)
1815, 16, 173bitr4i 292 . 2 (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1918eqriv 2618 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908   cuni 4407   ciun 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-uni 4408  df-iun 4492
This theorem is referenced by:  ituniiun  9196  tgidm  20708  txcmplem2  21368
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