[proofplan] An elementary embedding preserves and reflects truth of every formula with parameters from the domain, so the forward implication is immediate from the definition. Conversely, assume only one-way preservation for all formulas. Applying that assumption to the negation $\neg\varphi$ gives reflection of truth for $\varphi$, because failure of $\varphi$ is exactly truth of $\neg\varphi$ under the satisfaction clause for negation. Thus one-way preservation for all formulas is equivalent to full elementary preservation. [/proofplan]

[step:Derive one-way preservation from elementary equivalence on tuples] Assume that $f:M\to N$ is an elementary embedding. By definition, for every $k\in\mathbb N$, every $L$-formula $\varphi(x_1,\dots,x_k)$, and every tuple $a\in M^k$, \begin{align*} M\models \varphi(a)\iff N\models \varphi(f(a)). \end{align*} The implication \begin{align*} M\models \varphi(a)\implies N\models \varphi(f(a)) \end{align*} is therefore one direction of this equivalence. [/step]

[step:Use preservation of negations to recover reflection] Assume conversely that, for every $k\in\mathbb N$, every $L$-formula $\psi(x_1,\dots,x_k)$, and every tuple $b\in M^k$, \begin{align*} M\models \psi(b)\implies N\models \psi(f(b)). \end{align*} Fix $k\in\mathbb N$, an $L$-formula $\varphi(x_1,\dots,x_k)$, and a tuple $a\in M^k$. The assumed implication applied to $\psi=\varphi$ gives \begin{align*} M\models \varphi(a)\implies N\models \varphi(f(a)). \end{align*} It remains to prove the reverse implication. Suppose that $N\models \varphi(f(a))$. If $M\not\models \varphi(a)$, then the satisfaction clause for negation gives \begin{align*} M\models \neg\varphi(a). \end{align*} Since $\neg\varphi(x_1,\dots,x_k)$ is also an $L$-formula, the assumed one-way preservation applied to $\psi=\neg\varphi$ gives \begin{align*} N\models \neg\varphi(f(a)). \end{align*} Again by the satisfaction clause for negation, this means $N\not\models \varphi(f(a))$, contradicting the choice of $a$ with $N\models \varphi(f(a))$. Hence $M\models \varphi(a)$. Therefore, for every $k\in\mathbb N$, every $L$-formula $\varphi(x_1,\dots,x_k)$, and every tuple $a\in M^k$, \begin{align*} M\models \varphi(a)\iff N\models \varphi(f(a)). \end{align*} [/step]

[step:Identify the resulting equivalence with elementarity] The last displayed equivalence is exactly the defining condition that the embedding $f:M\to N$ is elementary: every formula with parameters from $M$ has the same truth value in $M$ at $a$ as in $N$ at the image tuple $f(a)$. Thus the assumed one-way preservation property implies that $f$ is an elementary embedding, completing the proof. [/step]