Theorem dif32 4193

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 4053	. . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
2	1	difeq2i 4020	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (𝐶 ∪ 𝐵))
3		difun1 4190	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶)
4		difun1 4190	. 2 ⊢ (𝐴 ∖ (𝐶 ∪ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
5	2, 3, 4	3eqtr3i 2767	1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)

Syntax hints:   = wceq 1543   ∖ cdif 3850   ∪ cun 3851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860

This theorem is referenced by:  difdifdir  4389  difsnen  8705  nbupgruvtxres  27449  poimirlem25  35488