From this resource, the writer starts with a linear model:
$$ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_m x_m $$
and then makes the RHS sigmoidal. This must then make the LHS sigmoidal to preserve the equality.
$$\implies S(y) = S(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_m x_m) = \frac{1}{1+ \exp^{-(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... +\beta_m x_m)}} $$
But he instead writes
$$ P(y=1) = S(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_m x_m) $$
Could someone explain the implication that $S(y) = P(y=1)$?