The real formal procedure:

dy/dx = g(x)f(y)

Let h(y) = 1/f(y)

=> dy/dx = g(x)/h(y)

=> h(y) dy/dx = g(x)

Now, we integrate both sides,

int h(y) dy/dx dx = int g(x) dx

But the left hand side is the same as

int h(y) dy by substitution rule of integration.

Therefore,

int h(y) dy = int g(x) dx

Proceed with solving now, no abuse since the substitution rule is provable. QED