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Theorem iuncom 3637
Description: Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
iuncom x A y B 𝐶 = y B x A 𝐶
Distinct variable groups:   y,A   x,B   x,y
Allowed substitution hints:   A(x)   B(y)   𝐶(x,y)

Proof of Theorem iuncom
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 rexcom 2452 . . . 4 (x A y B z 𝐶y B x A z 𝐶)
2 eliun 3635 . . . . 5 (z y B 𝐶y B z 𝐶)
32rexbii 2309 . . . 4 (x A z y B 𝐶x A y B z 𝐶)
4 eliun 3635 . . . . 5 (z x A 𝐶x A z 𝐶)
54rexbii 2309 . . . 4 (y B z x A 𝐶y B x A z 𝐶)
61, 3, 53bitr4i 201 . . 3 (x A z y B 𝐶y B z x A 𝐶)
7 eliun 3635 . . 3 (z x A y B 𝐶x A z y B 𝐶)
8 eliun 3635 . . 3 (z y B x A 𝐶y B z x A 𝐶)
96, 7, 83bitr4i 201 . 2 (z x A y B 𝐶z y B x A 𝐶)
109eqriv 2019 1 x A y B 𝐶 = y B x A 𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1228   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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