This discussion is clearly meant to be carried out in the metalanguage, and we must forgive his use of an object-language quantifier where he should be using the quantificational devices of the metalanguage, and his use of " ", which appears to be a variable of the object-language, when he seems to be using "x " schematically for any wff with no bound variables and "x " as its only free variable. What he means is that a quantified proposition is true if and only if every proposition got by replacing its variable with an individual constant is true, or more precisely:
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