Theorem equncom 3272

Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. (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 3271	. 2 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
2	1	eqeq2i 2181	1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

Syntax hints:   ↔ wb 104   = wceq 1348   ∪ cun 3119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125

This theorem is referenced by:  equncomi  3273