Exogeneity assumption in OLS

Question

Recently I read chapter 1 of Hayashi's Econometrics, the one explaining the finite-sample properties of OLS, and I had a doubt regarding the implications of the strict exogeneity assumption. Let's say we have the following relation:

$$y_i = \beta_o + \beta_1x_{i1} + \beta_2x_{i2} + \epsilon_i $$

where $y_i$ is the i-th observation of the dependent variable, $(x_{i1},x_{i2})$ the i-th observation of the 2 regressors, $\epsilon_i$ the error term and $\beta_0$, $\beta_1$, $\beta_2$ the parameters.

According to the strict exogeneity assumption $$E[\epsilon_i|x_{i1}] = 0$$ (i.e. the expected value of the error term is zero coditional on the first regressor) and $$E[\epsilon_i|x_{i2}] = 0$$ (i.e. the expected value of the error term is zero coditional on the second regressor). My question is whether, from these two facts, it follows that $$E[\epsilon_i|x_{i1},x_{i2}] = 0 $$ (i.e. the expected value of the error term is zero conditional on both the regressors).

I tried to verify whether the last equality holds by writing the following code in Python:

import matplotlib.pyplot as plt
import numpy as np

mean = [0, 0, 0]
cov = [[1, 0.4, 0], [0.4, 1, 0], [0, 0, 1]]

#Draws from a multivariate normal distribution (where corr.coeff(x1,x2) = 0.4, corr.coeff(x1,error) = 0 and corr.coeff(x2,error) = 0)
X = np.random.multivariate_normal(mean, cov, 100000)
x1 = X[:,0]
x2 = X[:,1]
error = X[:,2]

#I calculate the expected value of the error term conditional on x1 assuming values between -2 and -1 and on x2 assuming values between -2 and -1.
points_interval = []
for i in range(0,100000):
    if -2<=X[i,0]<=-1 and -2<=X[i,1]<=-1:
        points_interval.append(X[i,:])
mean_error_points_interval = np.mean(np.asarray(points_interval)[:,2])
#The expected value of the error conditional on x1 and x2 is approximately 0 (for different values of x1 and x2).

This led me to think that $$E[\epsilon_i|x_{i1},x_{i2}] = 0 $$ holds. I just can't see whether this fact immediately follows from the conditional EV of the error being independent from $x_1$, on the one hand, and from $x_2$, on the other. I'd be very grateful if someone could provide a formal derivation.