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Theorem iuncom 3975
 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 2772 . . . 4 (x A y B z Cy B x A z C)
2 eliun 3973 . . . . 5 (z y B Cy B z C)
32rexbii 2639 . . . 4 (x A z y B Cx A y B z C)
4 eliun 3973 . . . . 5 (z x A Cx A z C)
54rexbii 2639 . . . 4 (y B z x A Cy B x A z C)
61, 3, 53bitr4i 268 . . 3 (x A z y B Cy B z x A C)
7 eliun 3973 . . 3 (z x A y B Cx A z y B C)
8 eliun 3973 . . 3 (z y B x A Cy B z x A C)
96, 7, 83bitr4i 268 . 2 (z x A y B Cz y B x A C)
109eqriv 2350 1 x A y B C = y B x A C
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-iun 3971 This theorem is referenced by: (None)
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