Theorem un12 3422

Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.)

Assertion

Ref	Expression
un12	⊢ (A ∪ (B ∪ C)) = (B ∪ (A ∪ C))

Proof of Theorem un12

Step	Hyp	Ref	Expression
1		uncom 3409	. . 3 ⊢ (A ∪ B) = (B ∪ A)
2	1	uneq1i 3415	. 2 ⊢ ((A ∪ B) ∪ C) = ((B ∪ A) ∪ C)
3		unass 3421	. 2 ⊢ ((A ∪ B) ∪ C) = (A ∪ (B ∪ C))
4		unass 3421	. 2 ⊢ ((B ∪ A) ∪ C) = (B ∪ (A ∪ C))
5	2, 3, 4	3eqtr3i 2381	1 ⊢ (A ∪ (B ∪ C)) = (B ∪ (A ∪ C))

Syntax hints:   = 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:  un23  3423  un4  3424