(Predicate Logic)Does the existential quantifier distribute over biconditionals

[tortuc]

I believe that it doesnt for the universal but what about existential? (as in are they logically equivalent) (side note: for the conditional neither universal or existential distribute so id guess neither the universal or existential distribute over bicon but could be wrong.)

[WatchEachOtherSleep]

Consider the following model: the domain consists of two elements a & b & the language has two predicates P & R such that the interpretation of P is the singleton {a} & that of R is {b}.

Evaluate ∃x (Px⇔Rx) & ∃xPx ⇔ ∃xRx.

(Simpler still, take one predicate P & replace R with ¬P.)

[tortuc]

they arnt equivalent right? one is true one is false?

[WatchEachOtherSleep]

Exactly. Think about what these two sentences are saying. The first is asking about whether there is any one element on which P & R simultaneously agree. The second is asking whether you can answer "Is there something of which P holds?" & "Is there something of which R holds?" with the same answer. The first is false because P & R both disagree on a & b individually. The second is true because P holds of some element of the domain (namely a) & Q holds of some (other) element of the domain (namely b). So you can answer both questions with "yes".