Theorem dif32 3494

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 3373	. . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
2	1	difeq2i 3344	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (𝐶 ∪ 𝐵))
3		difun1 3491	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶)
4		difun1 3491	. 2 ⊢ (𝐴 ∖ (𝐶 ∪ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
5	2, 3, 4	3eqtr3i 2267	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)

Syntax hints:   = wceq 1402   ∖ cdif 3217   ∪ cun 3218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226

This theorem is referenced by:  difdifdirss  3609