Theorem equncom 4139

Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4139 was automatically derived from equncomVD 44845 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 4138	. 2 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
2	1	eqeq2i 2747	1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

Syntax hints:   ↔ wb 206   = wceq 1539   ∪ cun 3929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-un 3936

This theorem is referenced by:  equncomi  4140  equncomiVD  44846