for all x for all y (x=y)
This is true when for every element in the domain, x = y is true.
equality holds true when LHS = RHS means if x value is = to y value then its true.
And only if LHS = RHS,then the Predicate is TRUE. and that is possible when both LHS and RHS has same value. That is single element.
The moment we take 2 values they both should be same or both should be differrent.
Case 1 : If both are same then LHS = RHS means same element repeated twice taken as single element in the domain (which is a set)
Case 2 : if both are differrent LHS $\neq$ RHS, that means there are 2 differrent elements in the domain and the predicate becomes false in this case.
So, The conclusion is When the Universe contains only 1 element,then the Predicate is always true.