# semantic properties of substitutability

In addition, we also have the following semantic properties on substitutability:

1. 1.

If $x,y$ are respectively free for $z$ in $A$, and do not occur free in $A$, then

 $\models\exists xA[x/z]\leftrightarrow\exists yA[y/z]\qquad\mbox{and}\qquad% \models\forall xA[x/z]\leftrightarrow\forall yA[y/z]$
2. 2.
• $\models(t_{1}=t_{2})\to(s[t_{1}/x]=s[t_{2}/x])$

• $\models(t_{1}=t_{2})\to(A[t_{1}/x]=A[t_{2}/x])$

3. 3.

Every wff $A$ is equivalent to a wff $A^{\prime}$ in the sense $\models A\leftrightarrow A^{\prime}$ such that $A^{\prime}$ contains no variables that are both free and bound.

Title semantic properties of substitutability SemanticPropertiesOfSubstitutability 2013-03-22 19:36:00 2013-03-22 19:36:00 CWoo (3771) CWoo (3771) 6 CWoo (3771) Result msc 03B10 msc 03B05