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Theorem 2iunin 3714
Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
2iunin x A y B (𝐶𝐷) = ( x A 𝐶 y B 𝐷)
Distinct variable groups:   x,B   y,𝐶   x,𝐷   x,y
Allowed substitution hints:   A(x,y)   B(y)   𝐶(x)   𝐷(y)

Proof of Theorem 2iunin
StepHypRef Expression
1 iunin2 3711 . . . 4 y B (𝐶𝐷) = (𝐶 y B 𝐷)
21a1i 9 . . 3 (x A y B (𝐶𝐷) = (𝐶 y B 𝐷))
32iuneq2i 3666 . 2 x A y B (𝐶𝐷) = x A (𝐶 y B 𝐷)
4 iunin1 3712 . 2 x A (𝐶 y B 𝐷) = ( x A 𝐶 y B 𝐷)
53, 4eqtri 2057 1 x A y B (𝐶𝐷) = ( x A 𝐶 y B 𝐷)
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  cin 2910   ciun 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650
This theorem is referenced by: (None)
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