I wish to formulate a proof that if $x+y = x+z$ and $xy$ = $xz$ then $y=z$. I'm just beginning my study of Boolean algebra, but is $y=z$ not self evident from the stated equations?
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$x$ is either true or false. In either case you can derive $y=z$ from one of the equations.
Use absorption and distributivity: $$y=y(x+y)=y(x+z)=yx+yz=zx+yz=z(x+y)=z(x+z)=z.$$