[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Prove that if

[[math]] P(A|C) \geq P(B|C) \mbox{\,\,and\,\,} P(A|\tilde C) \geq P(B|\tilde C)\ , [[/math]]

then [math]P(A) \geq P(B)[/math].