Theorem dif32 3422

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 3303	. . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
2	1	difeq2i 3274	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (𝐶 ∪ 𝐵))
3		difun1 3419	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶)
4		difun1 3419	. 2 ⊢ (𝐴 ∖ (𝐶 ∪ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
5	2, 3, 4	3eqtr3i 2222	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)

Syntax hints:   = wceq 1364   ∖ cdif 3150   ∪ cun 3151

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159

This theorem is referenced by:  difdifdirss  3531