As in the title: does the existential quantifier distribute over equivalence?
Is this true: $\exists_{x} \left( \phi \left( x \right) \Leftrightarrow \psi \left( x \right) \right) \Leftrightarrow \left( \exists_{x} \phi \left( x \right) \Leftrightarrow \exists_{x} \psi \left( x \right) \right)$ ?
Or at least implication from left to right side?
No. They are not equivalent, and in fact neither side implies the other.
Counterexample 1: have two objects $a$ and $b$, with $a$ having property $\phi$ but not $\psi$, and $b$ having property $\psi$ but not $\phi$. Then the right side is true, but the left side is false.
Counterexample 2: have two objects $a$ and $b$, with $a$ having property $\phi$ but not $\psi$, and $b$ having neither of those two properties. Then the left side is true, but the right side is false.
