Metamath Proof Explorer

Theorem 2ralbii

Description: Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004)

Ref Expression
Hypothesis ralbii.1 
|- ( ph <-> ps )
Assertion 2ralbii 
|- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps )

Proof

Step Hyp Ref Expression
1 ralbii.1 
 |-  ( ph <-> ps )
2 1 ralbii 
 |-  ( A. y e. B ph <-> A. y e. B ps )
3 2 ralbii 
 |-  ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps )