Theorem dif32 3517

Description: Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)

Assertion

Ref	Expression
dif32	⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∖ B)

Proof of Theorem dif32

Step	Hyp	Ref	Expression
1		uncom 3408	. . 3 ⊢ (B ∪ C) = (C ∪ B)
2	1	difeq2i 3382	. 2 ⊢ (A ∖ (B ∪ C)) = (A ∖ (C ∪ B))
3		difun1 3514	. 2 ⊢ (A ∖ (B ∪ C)) = ((A ∖ B) ∖ C)
4		difun1 3514	. 2 ⊢ (A ∖ (C ∪ B)) = ((A ∖ C) ∖ B)
5	2, 3, 4	3eqtr3i 2381	1 ⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∖ B)

Syntax hints:   = wceq 1642   ∖ cdif 3206   ∪ cun 3207

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215

This theorem is referenced by:  difdifdir  3637