Theorem inindir 2231

Description: Intersection distributes over itself.

Assertion

Ref	Expression
inindir	|- ((A i^i B) i^i C) = ((A i^i C) i^i (B i^i C))

Proof of Theorem inindir

Step	Hyp	Ref	Expression
1		inidm 2225	. . 3 |- (C i^i C) = C
2	1	ineq2i 2217	. 2 |- ((A i^i B) i^i (C i^i C)) = ((A i^i B) i^i C)
3		in4 2229	. 2 |- ((A i^i B) i^i (C i^i C)) = ((A i^i C) i^i (B i^i C))
4	2, 3	eqtr3 1500	1 |- ((A i^i B) i^i C) = ((A i^i C) i^i (B i^i C))

Syntax hints:   = wceq 958   i^i cin 2049

This theorem is referenced by:  difindir 2263  subtop 7643  mdslmd1lem1 10247  mdslmd1lem2 10248  mddmdin0 10353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054

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