Theorem ineq12i

Description: Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004) (Proof shortened by Eric Schmidt, 26-Jan-2007)

	Ref	Expression
Hypotheses	ineq1i.1	⊢ 𝐴 = 𝐵
	ineq12i.2	⊢ 𝐶 = 𝐷
Assertion	ineq12i	⊢ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		ineq1i.1	⊢ 𝐴 = 𝐵
2		ineq12i.2	⊢ 𝐶 = 𝐷
3		ineq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 )