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I have to prove this using real world examples and I don't know how. I tried to do this with nicotine, but the unique existential quantifier confuses me.

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    Why are these equivalent? What if $A(x)$ was the statement $x=1$? – 2017-02-05
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    $A(x)$ could be $x$ is queen of England or $x$ is a natural satellite of Earth $x$ is my wife (if bigamy is unlawful)... – 2017-02-05
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    @benji if $A(x)\cong x=1$ then the statement just assert that there a unique 1, both parts says so. If $\exists xA(x)$ then there is a 1, it may not be unique but at least we can confirm existence, but if $y=1$ and $z=1$ then, obviously, $z=y$. So it is unique. – 2017-02-05
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    @sufronausea Thanks for the clarification. After reading the answer, I realize I completely misunderstood what the symbol meant. I thought it meant "there doesn't exist x such that..." but seeing that it means there is exactly one, then this makes perfect sense to me. – 2017-02-05
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    Hard to to tell you how to prove it without knowing what def'n of "$\exists!$" you are given, as the RHS would, to me, be a def'n of it. – 2017-02-05

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Ok, so let's say that $A(x)$ means $x$ is a perfect milkshake.

then $\exists ! x A(x)$ states that there is exactly one perfect milkshake.

The $\exists x A(x) \land \forall y \forall z ((A(y) \land A(z)) \rightarrow y=z)$ statement states two things:

$\exists x A(x)$ ... This means that there is at least one perfect milkshake

$\forall y \forall z ((A(y) \land A(z)) \rightarrow y=z)$ ... This means that if you ever have two perfect milkshakes, then they are really just one and the same ... meaning that you can never have two different perfect milkshakes, and thus you can have only at most one perfect milkshake.

But if you have at least one perfect milkshake, and you can't have more than one, then you must have exactly one!

So, the two statements say the same thing!