Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  equncom Structured version   Visualization version   GIF version

Theorem equncom 3720
 Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 3720 was automatically derived from equncomVD 38126 using the tools program translatewithout_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 3719 . 2 (𝐵𝐶) = (𝐶𝐵)
21eqeq2i 2622 1 (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∪ cun 3538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545 This theorem is referenced by:  equncomi  3721  equncomiVD  38127
 Copyright terms: Public domain W3C validator