Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004)
Ref | Expression | ||
---|---|---|---|
Assertion | difindir | |- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inindir | |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) ) |
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2 | invdif | |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i B ) \ C ) |
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3 | invdif | |- ( A i^i ( _V \ C ) ) = ( A \ C ) |
|
4 | invdif | |- ( B i^i ( _V \ C ) ) = ( B \ C ) |
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5 | 3 4 | ineq12i | |- ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) i^i ( B \ C ) ) |
6 | 1 2 5 | 3eqtr3i | |- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) ) |