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Theorem iunun 3708
Description: Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunun x A (B𝐶) = ( x A B x A 𝐶)

Proof of Theorem iunun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 r19.43 2446 . . . 4 (x A (y B y 𝐶) ↔ (x A y B x A y 𝐶))
2 elun 3061 . . . . 5 (y (B𝐶) ↔ (y B y 𝐶))
32rexbii 2309 . . . 4 (x A y (B𝐶) ↔ x A (y B y 𝐶))
4 eliun 3635 . . . . 5 (y x A Bx A y B)
5 eliun 3635 . . . . 5 (y x A 𝐶x A y 𝐶)
64, 5orbi12i 668 . . . 4 ((y x A B y x A 𝐶) ↔ (x A y B x A y 𝐶))
71, 3, 63bitr4i 201 . . 3 (x A y (B𝐶) ↔ (y x A B y x A 𝐶))
8 eliun 3635 . . 3 (y x A (B𝐶) ↔ x A y (B𝐶))
9 elun 3061 . . 3 (y ( x A B x A 𝐶) ↔ (y x A B y x A 𝐶))
107, 8, 93bitr4i 201 . 2 (y x A (B𝐶) ↔ y ( x A B x A 𝐶))
1110eqriv 2019 1 x A (B𝐶) = ( x A B x A 𝐶)
Colors of variables: wff set class
Syntax hints:   wo 616   = wceq 1228   wcel 1374  wrex 2285  cun 2892   ciun 3631
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-iun 3633
This theorem is referenced by: (None)
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