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**1. Homework Statement**

Determine whether [tex]\forall\,x\,(P(x)\,\longleftrightarrow\,Q(x))[/tex] and [tex]\forall\,x\,P(x)\,\longleftrightarrow\,\forall\,x\,Q(x)[/tex] are logically equivalent. Justify your answer.

**2. Homework Equations**

[tex]P\,\longleftrightarrow\,Q[/tex] is only TRUE when both P and Q are TRUE or FALSE.

**3. The Attempt at a Solution**

No, I don't think the two statements are logically equivalent, but I have trouble trying to "justify" my answer.

Set P and Q as always TRUE.

Both statements are equivalent, but if you set P and Q to always FALSE, then the statements are no longer equivalent.

Does this seem logical