We have a simple linear regression model. Our assumptions are:

$Y\_i =\beta\_0+\beta\_1X\_i+ \varepsilon\_i $, $i=1, \cdots, n$

$\varepsilon\_i \sim N(0, \sigma^2)$

$Var(\varepsilon\_i|X\_i=x)=\sigma^2$ 

$\varepsilon\_1, \cdots, \varepsilon\_n$ are mutually independent.

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Are these hypothesis enough to claim that $\varepsilon\_i|X\_i=x \sim N(0, \sigma^2)$?