Bi-implication theorem proving

Question

While proving a theorem, i came across a situation like as follows

(P has a property) $\leftrightarrow $ $(x=y)$

(P has a property) $\leftrightarrow $ $(y=z)$

Now can i infer the following fact from the above two facts ?

(P has a property) $\leftrightarrow $ $(x=z)$

$\leftrightarrow $ stands for Bi-conditional(if and only if)

Thanks in advance :)

Answer 1

The implication $\rightarrow$ is clearly correct, because if $P$ has the property, then $x = y = z$. The implication $\leftarrow$ doesn't hold in general though. For example, let the property of $P$ be "$x=y=z$ holds". Then in the case $x = z = 0$, $y = 1$ the statement $x = z$ is true but the property of $P$ is not satisfied.

Answer 2

You can infer the left-to-right component of your third biconditional, but not the right-to-left. If $x\neq y$ and $z\neq y$, you can still have $x=z$ while $P$ is clearly false.