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Theorem iunxiun 3710
Description: Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
iunxiun x y A B𝐶 = y A x B 𝐶
Distinct variable groups:   x,y   x,A   y,𝐶
Allowed substitution hints:   A(y)   B(x,y)   𝐶(x)

Proof of Theorem iunxiun
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eliun 3635 . . . . . . . 8 (x y A By A x B)
21anbi1i 434 . . . . . . 7 ((x y A B z 𝐶) ↔ (y A x B z 𝐶))
3 r19.41v 2444 . . . . . . 7 (y A (x B z 𝐶) ↔ (y A x B z 𝐶))
42, 3bitr4i 176 . . . . . 6 ((x y A B z 𝐶) ↔ y A (x B z 𝐶))
54exbii 1478 . . . . 5 (x(x y A B z 𝐶) ↔ xy A (x B z 𝐶))
6 rexcom4 2554 . . . . 5 (y A x(x B z 𝐶) ↔ xy A (x B z 𝐶))
75, 6bitr4i 176 . . . 4 (x(x y A B z 𝐶) ↔ y A x(x B z 𝐶))
8 df-rex 2290 . . . 4 (x y A Bz 𝐶x(x y A B z 𝐶))
9 eliun 3635 . . . . . 6 (z x B 𝐶x B z 𝐶)
10 df-rex 2290 . . . . . 6 (x B z 𝐶x(x B z 𝐶))
119, 10bitri 173 . . . . 5 (z x B 𝐶x(x B z 𝐶))
1211rexbii 2309 . . . 4 (y A z x B 𝐶y A x(x B z 𝐶))
137, 8, 123bitr4i 201 . . 3 (x y A Bz 𝐶y A z x B 𝐶)
14 eliun 3635 . . 3 (z x y A B𝐶x y A Bz 𝐶)
15 eliun 3635 . . 3 (z y A x B 𝐶y A z x B 𝐶)
1613, 14, 153bitr4i 201 . 2 (z x y A B𝐶z y A x B 𝐶)
1716eqriv 2019 1 x y A B𝐶 = y A x B 𝐶
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wex 1362   wcel 1374  wrex 2285   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-iun 3633
This theorem is referenced by: (None)
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