Theorem inteq

Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999)

	Ref	Expression
Assertion	inteq	$$⊢A=B\to \bigcap A=\bigcap B$$

Proof

Step	Hyp	Ref	Expression
1		raleq	$$⊢A=B\to \left(\forall y\in Ax\in y\leftrightarrow \forall y\in Bx\in y\right)$$
2	1	abbidv	$$⊢A=B\to \left\{x|\forall y\in Ax\in y\right\}=\left\{x|\forall y\in Bx\in y\right\}$$
3		dfint2	$$⊢\bigcap A=\left\{x|\forall y\in Ax\in y\right\}$$
4		dfint2	$$⊢\bigcap B=\left\{x|\forall y\in Bx\in y\right\}$$
5	2 3 4	3eqtr4g	$$⊢A=B\to \bigcap A=\bigcap B$$