# Time invariance from convolution integral

I wanted to prove, time invariance property of system by convolution integral(i.e equal time shift in input result in equal shift in output) but as calculation shown in figure, I got unexpected result, so what I'm missing?

If your delayed input is $$\hat{x}(t)=x(t-\alpha)$$, then $$\hat{x}(t-\tau)=x(t-\tau-\alpha)$$, and not $$x(t-\tau+\alpha)$$. From here everything will work out as expected.