Theorem difindi 1683

Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29.

Assertion

Ref	Expression
difindi	|- (A \ (B i^i C)) = ((A \ B) u. (A \ C))

Proof of Theorem difindi

Step	Hyp	Ref	Expression
1		dfin3 1672	. . 3 |- (B i^i C) = (V \ ((V \ B) u. (V \ C)))
2	1	difeq2i 1585	. 2 |- (A \ (B i^i C)) = (A \ (V \ ((V \ B) u. (V \ C))))
3		indi 1676	. . 3 |- (A i^i ((V \ B) u. (V \ C))) = ((A i^i (V \ B)) u. (A i^i (V \ C)))
4		dfin2 1669	. . 3 |- (A i^i ((V \ B) u. (V \ C))) = (A \ (V \ ((V \ B) u. (V \ C))))
5		invdif 1674	. . . 4 |- (A i^i (V \ B)) = (A \ B)
6		invdif 1674	. . . 4 |- (A i^i (V \ C)) = (A \ C)
7	5, 6	uneq12i 1609	. . 3 |- ((A i^i (V \ B)) u. (A i^i (V \ C))) = ((A \ B) u. (A \ C))
8	3, 4, 7	3eqtr3 1124	. 2 |- (A \ (V \ ((V \ B) u. (V \ C)))) = ((A \ B) u. (A \ C))
9	2, 8	eqtr 1119	1 |- (A \ (B i^i C)) = ((A \ B) u. (A \ C))

Syntax hints:   = wceq 1091  Vcvv 1348   \ cdif 1484   u. cun 1485   i^i cin 1486

This theorem is referenced by:  indm 1686

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

metamath.org