Theorem cnvin

Description: Distributive law for converse over intersection. Theorem 15 of Suppes p. 62. (Contributed by NM, 25-Mar-1998) (Revised by Mario Carneiro, 26-Jun-2014)

	Ref	Expression
Assertion	cnvin	⊢ ◡ ( 𝐴 ∩ 𝐵 ) = ( ◡ 𝐴 ∩ ◡ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		cnvdif	⊢ ◡ ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) ) = ( ◡ 𝐴 ∖ ◡ ( 𝐴 ∖ 𝐵 ) )
2		cnvdif	⊢ ◡ ( 𝐴 ∖ 𝐵 ) = ( ◡ 𝐴 ∖ ◡ 𝐵 )
3	2	difeq2i	⊢ ( ◡ 𝐴 ∖ ◡ ( 𝐴 ∖ 𝐵 ) ) = ( ◡ 𝐴 ∖ ( ◡ 𝐴 ∖ ◡ 𝐵 ) )
4	1 3	eqtri	⊢ ◡ ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) ) = ( ◡ 𝐴 ∖ ( ◡ 𝐴 ∖ ◡ 𝐵 ) )
5		dfin4	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) )
6	5	cnveqi	⊢ ◡ ( 𝐴 ∩ 𝐵 ) = ◡ ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) )
7		dfin4	⊢ ( ◡ 𝐴 ∩ ◡ 𝐵 ) = ( ◡ 𝐴 ∖ ( ◡ 𝐴 ∖ ◡ 𝐵 ) )
8	4 6 7	3eqtr4i	⊢ ◡ ( 𝐴 ∩ 𝐵 ) = ( ◡ 𝐴 ∩ ◡ 𝐵 )