Metamath Proof Explorer

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

Proof

Step Hyp Ref Expression
1 ralel 
 |-  A. x e. B x e. B
2 raleq 
 |-  ( A = B -> ( A. x e. A x e. B <-> A. x e. B x e. B ) )
3 1 2 mpbiri 
 |-  ( A = B -> A. x e. A x e. B )