Conditional mutual information

I have three RVs X, Y, and T. Is the following equation true? I(X ; Y|T) = I(Y ; X|T) Can we express the conditional mutual information as: (X;Y|T) = I(X;Y) - I(X;Y;T) ?

Yes, $$I(X ; Y|T)=H(X|T)+H(Y|T)-H(X,Y |T)=I(Y ; X|T)$$.
Note that $$H(A|B)=H(A)-I(A;B)$$, so we can write $$H(X|T)=H(X)-I(X;T)$$ and $$H(Y|T)=H(Y)-I(Y;T)$$. Similarly for $$H(X,Y | T)$$.
Now we can collect $$H(X), H(Y)$$ and $$H(X,Y)$$ from first terms of all three to write $$H(X)+H(Y)-H(X,Y)=I(X;Y)$$ i.e. $$I(X ; Y|T)=I(X;Y)-(I(X;T)+I(Y;T)-I(X,Y; T) )=I(X;Y)-I(X;Y;T)$$ Like @mhdadk pointed out, all identities are in mi and conditional mutual information