Equivalent Replacement (Propositional Logic)

Definition

Equivalent Replacement

Let $A[B]$ denote a formula $A$ with one fixed occurrence of a subformula $B$, identified by a position in $A$, and let $A[B\to B^{\prime }]$ be the formula obtained by replacing that occurrence with $B^{\prime }$.

For every interpretation $I$,

$$I\models B_1\leftrightarrow B_2⟹I\models A[B_1]\leftrightarrow A[B_2].$$

Two subformulas with the same truth value under $I$ are interchangeable at a fixed occurrence: replacing one by the other leaves the truth value of the whole formula unchanged under $I$.

Intuition

Compositionality

The truth value of a formula depends on the truth values of its immediate parts. Therefore, if one occurrence of a subformula is replaced by another with the same value under the same interpretation, the value of the whole formula stays the same.