Theorem raleleq

Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020) (Proof shortened by Wolf Lammen, 18-Jul-2025)

	Ref	Expression
Assertion	raleleq	$$⊢A=B\to \forall x\in Ax\in B$$

Proof

Step	Hyp	Ref	Expression
1		ralel	$$⊢\forall x\in Bx\in B$$
2		raleq	$$⊢A=B\to \left(\forall x\in Ax\in B\leftrightarrow \forall x\in Bx\in B\right)$$
3	1 2	mpbiri	$$⊢A=B\to \forall x\in Ax\in B$$