Theorem dif23 1688

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 1604	. . 3 |- (B u. C) = (C u. B)
2	1	difeq2i 1585	. 2 |- (A \ (B u. C)) = (A \ (C u. B))
3		difun1 1687	. 2 |- (A \ (B u. C)) = ((A \ B) \ C)
4		difun1 1687	. 2 |- (A \ (C u. B)) = ((A \ C) \ B)
5	2, 3, 4	3eqtr3 1124	1 |- ((A \ B) \ C) = ((A \ C) \ B)

Syntax hints:   = wceq 1091   \ cdif 1484   u. cun 1485

This theorem is referenced by:  difdifdir 1765

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491

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