Metamath Proof Explorer


Theorem difundi

Description: Distributive law for class difference. Theorem 39 of Suppes p. 29. (Contributed by NM, 17-Aug-2004)

Ref Expression
Assertion difundi ( 𝐴 ∖ ( 𝐵𝐶 ) ) = ( ( 𝐴𝐵 ) ∩ ( 𝐴𝐶 ) )

Proof

Step Hyp Ref Expression
1 dfun3 ( 𝐵𝐶 ) = ( V ∖ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) )
2 1 difeq2i ( 𝐴 ∖ ( 𝐵𝐶 ) ) = ( 𝐴 ∖ ( V ∖ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) )
3 inindi ( 𝐴 ∩ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) = ( ( 𝐴 ∩ ( V ∖ 𝐵 ) ) ∩ ( 𝐴 ∩ ( V ∖ 𝐶 ) ) )
4 dfin2 ( 𝐴 ∩ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) = ( 𝐴 ∖ ( V ∖ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) )
5 invdif ( 𝐴 ∩ ( V ∖ 𝐵 ) ) = ( 𝐴𝐵 )
6 invdif ( 𝐴 ∩ ( V ∖ 𝐶 ) ) = ( 𝐴𝐶 )
7 5 6 ineq12i ( ( 𝐴 ∩ ( V ∖ 𝐵 ) ) ∩ ( 𝐴 ∩ ( V ∖ 𝐶 ) ) ) = ( ( 𝐴𝐵 ) ∩ ( 𝐴𝐶 ) )
8 3 4 7 3eqtr3i ( 𝐴 ∖ ( V ∖ ( ( V ∖ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) ) ) = ( ( 𝐴𝐵 ) ∩ ( 𝐴𝐶 ) )
9 2 8 eqtri ( 𝐴 ∖ ( 𝐵𝐶 ) ) = ( ( 𝐴𝐵 ) ∩ ( 𝐴𝐶 ) )