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Theorem equncom 4154
Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4154 was automatically derived from equncomVD 43931 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 4153 . 2 (𝐵𝐶) = (𝐶𝐵)
21eqeq2i 2745 1 (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  cun 3946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3953
This theorem is referenced by:  equncomi  4155  equncomiVD  43932
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