Theorem difindi 2259

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 2247	. . 3 |- (B i^i C) = (V \ ((V \ B) u. (V \ C)))
2	1	difeq2i 2156	. 2 |- (A \ (B i^i C)) = (A \ (V \ ((V \ B) u. (V \ C))))
3		indi 2251	. . 3 |- (A i^i ((V \ B) u. (V \ C))) = ((A i^i (V \ B)) u. (A i^i (V \ C)))
4		dfin2 2244	. . 3 |- (A i^i ((V \ B) u. (V \ C))) = (A \ (V \ ((V \ B) u. (V \ C))))
5		invdif 2249	. . . 4 |- (A i^i (V \ B)) = (A \ B)
6		invdif 2249	. . . 4 |- (A i^i (V \ C)) = (A \ C)
7	5, 6	uneq12i 2182	. . 3 |- ((A i^i (V \ B)) u. (A i^i (V \ C))) = ((A \ B) u. (A \ C))
8	3, 4, 7	3eqtr3 1503	. 2 |- (A \ (V \ ((V \ B) u. (V \ C)))) = ((A \ B) u. (A \ C))
9	2, 8	eqtr 1495	1 |- (A \ (B i^i C)) = ((A \ B) u. (A \ C))

Syntax hints:   = wceq 956  Vcvv 1811   \ cdif 2044   u. cun 2045   i^i cin 2046

This theorem is referenced by:  indm 2262  fctopOLD 7650  cctop 7652

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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