Let \( (X_1, X_2) \sim \mathcal{N}_2(\mu_1, \mu_2, \sigma^2_1, \sigma^2_2, \rho) \)

The CDF of \( X_1|X_2 > c \) is

\( \Pr\{X_1 \leq x|X_2 > c\} = \frac {\Pr\{X_1 \leq x, X_2 > c\}} {\Pr\{X_2 > c\}} \)

The denominator is just \( 1 - \Phi\left(\frac {c - \mu_2} {\sigma_2}\right) \), where \( \Phi \) is the standard normal CDF.

The numerator is \( \int_{-\infty}^x \int_c^{+\infty} f_{X_1, X_2}(u,v)dvdu\)

Differentiating the numerator with respect to \( x \), we have

\( \int_c^{+\infty} f_{X_1, X_2}(x,v)dv \)

\( = \int_c^{+\infty} \frac {1} {2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}} \)\(

\exp\left\{-\frac {1} {2(1-\rho^2)}\left[\frac {(x-\mu_1)^2} {\sigma_1^2}

+ \frac {(v-\mu_2)^2} {\sigma_2^2} - \frac {2\rho(x-\mu_1)(v-\mu_2)} {\sigma_1\sigma_2} \right]\right\}dv \)

\( = \frac {1} {2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}

\exp\left\{-\frac {1} {2(1-\rho^2)}\left[\frac {(x-\mu_1)^2} {\sigma_1^2}

- \frac {\rho^2(x-\mu_1)^2} {\sigma_1^2} \right]\right\}

\)

\( \times \int_c^{+\infty} \exp\left\{-\frac {1} {2(1-\rho^2)}

\left[\frac {v - \mu_2} {\sigma_2} - \frac {\rho(x - \mu_1)} {\sigma_1} \right]^2\right\}dv \)

\( = \frac {1} {2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}

\exp\left\{-\frac {1} {2(1-\rho^2)}\left[\frac {(x-\mu_1)^2} {\sigma_1^2}

- \frac {\rho^2(x-\mu_1)^2} {\sigma_1^2} \right]\right\} \)

\( \times \sqrt{2\pi\sigma_2^2(1-\rho^2)} \left[1 - \Phi\left(\frac {1} {\sigma_2\sqrt{1-\rho^2}}\left(c - \mu_2 - \frac {\rho\sigma_2(x-\mu_1)} {\sigma_1} \right)\right)\right]\)

\( = \frac {1} {\sqrt{2\pi\sigma_1^2}} \exp\left\{-\frac {1} {2} \frac {(x-\mu_1)^2} {\sigma_1^2} \right\} \times \left[1 - \Phi\left(\frac {1} {\sigma_2\sqrt{1-\rho^2}}\left(c - \mu_2 - \frac {\rho\sigma_2(x-\mu_1)} {\sigma_1} \right)\right)\right]\)

Combining you get the pdf of the conditional distribution. Sorry if there is any typo/mistakes