Originally Posted by

**Plato** For the first on I would expect:

$\displaystyle \left( {\exists x} \right)\left( {\forall y} \right)\left[ {\left( {T(x) \wedge B(x)} \right) \wedge \left( {\left( {T(y) \wedge B(y)} \right) \Rightarrow \left( {x = y} \right)} \right)} \right]$.

The second one could be:

$\displaystyle \left( {\exists x} \right)\left( {\exists y} \right)\left[ {\left( {T(x) \wedge C(x)} \right) \wedge \left( {\left( {T(y) \wedge C(y)} \right) \wedge \left( {x \not= y} \right)} \right)} \right]$