Martin's axiom is equivalent to the assertion that $$H_{\frak{c}}\prec_{\Sigma_1} V[G]$$ for all c.c.c. forcing extensions $V[G]$. In other words, $H_{\frak{c}}$ is existentially closed in all c.c.c. forcing extensions.

This is proved in Bagaria, Joan, A characterization of Martin’s axiom in terms of absoluteness, J. Symb. Log. 62, No. 2, 366-372 (1997). ZBL0883.03039.

This characterization leads naturally to the resurrection axioms, which I explored with Thomas Johnstone in several articles. The resurrection axiom for a class of forcing $\Gamma$ is the assertion that for every $\Gamma$ extension $V[g]$ there is further $\Gamma$ forcing $V[g][h]$ such that $$H_{\frak{c}}\prec H_{\frak{c}}^{V[g][h]}.$$ Thus, one attains full elementarity, at the cost of further forcing.

Martin's axiom is equivalent to the assertion that $$H_{\frak{c}}\prec_{\Sigma_1} V[G]$$ for all c.c.c. forcing extensions $V[G]$. In other words, $H_{\frak{c}}$ is existentially closed in all c.c.c. forcing extensions.

This is proved in Bagaria, Joan, A characterization of Martin’s axiom in terms of absoluteness, J. Symb. Log. 62, No. 2, 366-372 (1997). ZBL0883.03039. The characterization also appears independently, attributed to J. Stavi, in Stavi, Jonathan; Väänänen, Jouko, Reflection principles for the continuum, Zhang, Yi (ed.), Logic and algebra. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 302, 59-84 (2002). ZBL1013.03059.

This characterization leads naturally to the resurrection axioms, which I explored with Thomas Johnstone in several articles. The resurrection axiom for a class of forcing $\Gamma$ is the assertion that for every $\Gamma$ extension $V[g]$ there is further $\Gamma$ forcing $V[g][h]$ such that $$H_{\frak{c}}\prec H_{\frak{c}}^{V[g][h]}.$$ Thus, one attains full elementarity, at the cost of further forcing.