Use structural induction on formulas. Every term denotes the unique element. Every atomic predicate is true by hypothesis, and every equality is true because its two terms denote that same element.
If both immediate subformulas of an logical implication are true, their logical implication is true. A universal or existential quantification ranges over the singleton domain; its matrix is true for the only possible value by induction, so either quantified formula is true. These are all the formation cases in the reduced language. Hence
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