Theorem difindi 2257

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 2245	. . 3 |- (B i^i C) = (V \ ((V \ B) u. (V \ C)))
2	1	difeq2i 2154	. 2 |- (A \ (B i^i C)) = (A \ (V \ ((V \ B) u. (V \ C))))
3		indi 2249	. . 3 |- (A i^i ((V \ B) u. (V \ C))) = ((A i^i (V \ B)) u. (A i^i (V \ C)))
4		dfin2 2242	. . 3 |- (A i^i ((V \ B) u. (V \ C))) = (A \ (V \ ((V \ B) u. (V \ C))))
5		invdif 2247	. . . 4 |- (A i^i (V \ B)) = (A \ B)
6		invdif 2247	. . . 4 |- (A i^i (V \ C)) = (A \ C)
7	5, 6	uneq12i 2180	. . 3 |- ((A i^i (V \ B)) u. (A i^i (V \ C))) = ((A \ B) u. (A \ C))
8	3, 4, 7	3eqtr3 1502	. 2 |- (A \ (V \ ((V \ B) u. (V \ C)))) = ((A \ B) u. (A \ C))
9	2, 8	eqtr 1494	1 |- (A \ (B i^i C)) = ((A \ B) u. (A \ C))

Syntax hints:   = wceq 955  Vcvv 1809   \ cdif 2042   u. cun 2043   i^i cin 2044

This theorem is referenced by:  indm 2260  fctop 7629  cctop 7631

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  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 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1810  df-dif 2047  df-un 2048  df-in 2049

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