Theorem ineq1

Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993) (Proof shortened by SN, 20-Sep-2023)

	Ref	Expression
Assertion	ineq1	$$⊢A=B\to A\cap C=B\cap C$$

Proof

Step	Hyp	Ref	Expression
1		rabeq	$$⊢A=B\to \left\{x\in A|x\in C\right\}=\left\{x\in B|x\in C\right\}$$
2		dfin5	$$⊢A\cap C=\left\{x\in A|x\in C\right\}$$
3		dfin5	$$⊢B\cap C=\left\{x\in B|x\in C\right\}$$
4	1 2 3	3eqtr4g	$$⊢A=B\to A\cap C=B\cap C$$