Theorem equncom 3410

Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 3410 was automatically derived from equncomVD in set.mm 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 3409	. 2 ⊢ (B ∪ C) = (C ∪ B)
2	1	eqeq2i 2363	1 ⊢ (A = (B ∪ C) ↔ A = (C ∪ B))

Syntax hints:   ↔ wb 176   = wceq 1642   ∪ cun 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215

This theorem is referenced by:  equncomi  3411