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)

Ref Expression
Assertion raleleq 
|- ( A = B -> A. x e. A x e. B )

Proof

Step Hyp Ref Expression
1 eleq2 
 |-  ( A = B -> ( x e. A <-> x e. B ) )
2 1 biimpd 
 |-  ( A = B -> ( x e. A -> x e. B ) )
3 2 ralrimiv 
 |-  ( A = B -> A. x e. A x e. B )