Theorem equncom 4121

Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4121 was automatically derived from equncomVD 45467 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)

Assertion

Ref	Expression
equncom	⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

Proof of Theorem equncom

Step	Hyp	Ref	Expression
1		uncom 4120	. 2 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
2	1	eqeq2i 2782	1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

Syntax hints:   ↔ wb 209   = wceq 1567   ∪ cun 3911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918

This theorem is referenced by:  equncomi  4122  equncomiVD  45468