Odd initial condition makes heat equation odd

Question

Is my proof correct for saying that if the diffusion equation has an odd initial condition then the diffusion equation must be odd. I have this: $$\partial_{t} u - \partial_{xx} u = 0\\ u(x,0) = f(x)\\ So: u(x,t) = \int_{-\infty}^{\infty} \phi(x-y,t)f(y)dy\\ u(-x,t) = \frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty} e^{\frac{-(-x+y)^2}{4t}}(-f(-y))dy\\ \text{let } z = -y \text{ and } dz = -dy\\ = \frac{1}{\sqrt{4\pi t}}\int_{\infty}^{-\infty}e^{\frac{-(x-z)^2}{4t}}(f(z))dz\\ = \frac{-1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}e^{\frac{-(x-z)^2}{4t}}(f(z))dz\\ = -u(x,t)$$