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 -> A. x e. A x e. B )

Proof

Step	Hyp	Ref	Expression
1		dfcleq	|- ( A = B <-> A. x ( x e. A <-> x e. B ) )
2		biimp	|- ( ( x e. A <-> x e. B ) -> ( x e. A -> x e. B ) )
3	2	alimi	|- ( A. x ( x e. A <-> x e. B ) -> A. x ( x e. A -> x e. B ) )
4	1 3	sylbi	|- ( A = B -> A. x ( x e. A -> x e. B ) )
5		df-ral	|- ( A. x e. A x e. B <-> A. x ( x e. A -> x e. B ) )
6	4 5	sylibr	|- ( A = B -> A. x e. A x e. B )