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Theorem iunun 3725
 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 2462 . . . 4 (x A (y B y 𝐶) ↔ (x A y B x A y 𝐶))
2 elun 3078 . . . . 5 (y (B𝐶) ↔ (y B y 𝐶))
32rexbii 2325 . . . 4 (x A y (B𝐶) ↔ x A (y B y 𝐶))
4 eliun 3652 . . . . 5 (y x A Bx A y B)
5 eliun 3652 . . . . 5 (y x A 𝐶x A y 𝐶)
64, 5orbi12i 680 . . . 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 3652 . . 3 (y x A (B𝐶) ↔ x A y (B𝐶))
9 elun 3078 . . 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 2034 1 x A (B𝐶) = ( x A B x A 𝐶)
 Colors of variables: wff set class Syntax hints:   ∨ wo 628   = wceq 1242   ∈ wcel 1390  ∃wrex 2301   ∪ cun 2909  ∪ 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-bnd 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-un 2916  df-iun 3650 This theorem is referenced by: (None)
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