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Theorem equncom 4129
Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4129 was automatically derived from equncomVD 41082 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
Assertion
Ref Expression
equncom (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))

Proof of Theorem equncom
StepHypRef Expression
1 uncom 4128 . 2 (𝐵𝐶) = (𝐶𝐵)
21eqeq2i 2834 1 (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  cun 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-un 3940
This theorem is referenced by:  equncomi  4130  equncomiVD  41083
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