Theorem raleleq

Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020) Avoid ax-8 . (Revised by Wolf Lammen, 9-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		dfcleq	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$
2		biimp	$$⊢\left(x\in A\leftrightarrow x\in B\right)\to \left(x\in A\to x\in B\right)$$
3	2	alimi	$$⊢\forall x\left(x\in A\leftrightarrow x\in B\right)\to \forall x\left(x\in A\to x\in B\right)$$
4	1 3	sylbi	$$⊢A=B\to \forall x\left(x\in A\to x\in B\right)$$
5		df-ral	$$⊢\forall x\in Ax\in B\leftrightarrow \forall x\left(x\in A\to x\in B\right)$$
6	4 5	sylibr	$$⊢A=B\to \forall x\in Ax\in B$$