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question:

Prove if $A\subseteq$B Then A$\cap$B $\subseteq$ B$\cap$C 

solve:

A=B

(x$\in$ A)$\supseteq$ (x$\in$ B) $\implies${(x$\in$ A) $\wedge$ (x$\in$ C) , (x$\in$ B) $\wedge$ (x$\in$ C)}
$\implies$ (x$\in$ A) $\wedge$ (x$\in$ C) $\subseteq$ (x$\in$ B) $\wedge$ (x$\in$ C) $\implies$ A$\cap$B $\subseteq$ B$\cap$C