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Theorem iunin2 3694
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3684 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
iunin2 x A (B𝐶) = (B x A 𝐶)
Distinct variable group:   x,B
Allowed substitution hints:   A(x)   𝐶(x)

Proof of Theorem iunin2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 r19.42v 2445 . . . 4 (x A (y B y 𝐶) ↔ (y B x A y 𝐶))
2 elin 3103 . . . . 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 𝐶x A y 𝐶)
54anbi2i 433 . . . 4 ((y B y x A 𝐶) ↔ (y B x A y 𝐶))
61, 3, 53bitr4i 201 . . 3 (x A y (B𝐶) ↔ (y B y x A 𝐶))
7 eliun 3635 . . 3 (y x A (B𝐶) ↔ x A y (B𝐶))
8 elin 3103 . . 3 (y (B x A 𝐶) ↔ (y B y x A 𝐶))
96, 7, 83bitr4i 201 . 2 (y x A (B𝐶) ↔ y (B x A 𝐶))
109eqriv 2019 1 x A (B𝐶) = (B x A 𝐶)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1228   ∈ wcel 1374  ∃wrex 2285   ∩ cin 2893  ∪ 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-in 2901  df-iun 3633 This theorem is referenced by:  iunin1  3695  2iunin  3697  resiun1  4557  resiun2  4558
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