Theorem dif23 2264

Description: Swap second and third argument of double difference.

Assertion

Ref	Expression
dif23	|- ((A \ B) \ C) = ((A \ C) \ B)

Proof of Theorem dif23

Step	Hyp	Ref	Expression
1		uncom 2176	. . 3 |- (B u. C) = (C u. B)
2	1	difeq2i 2156	. 2 |- (A \ (B u. C)) = (A \ (C u. B))
3		difun1 2263	. 2 |- (A \ (B u. C)) = ((A \ B) \ C)
4		difun1 2263	. 2 |- (A \ (C u. B)) = ((A \ C) \ B)
5	2, 3, 4	3eqtr3 1503	1 |- ((A \ B) \ C) = ((A \ C) \ B)

Syntax hints:   = wceq 956   \ cdif 2044   u. cun 2045

This theorem is referenced by:  difdifdir 2346

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-un 2050  df-in 2051

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