The principle that one may replace a subformula by another subformula to which it is logically equivalent (i.e., φ ↔ ψ) within a larger formula without changing the overall logical value, provided contextual constraints are respected.

Principle

Whenever two formulas are provably equivalent or semantically equivalent in the given logic, occurrences of one may be uniformly substituted by the other in extensional contexts, preserving truth and provability.

Demonstration

If ¬¬p is provably equivalent to p in the logic at hand, then in any larger formula one may replace an occurrence of ¬¬p by p (for example turn ¬¬p ∨ q into p ∨ q) without changing logical consequence.

Misapplication

Replacing equivalents inside intensional or context-sensitive operators (such as belief, knowledge, modality, or within scopes that affect binding) where equivalence does not preserve meaning can invalidate arguments; also substituting formulas equivalent only under additional assumptions is unsafe.

Consequence

Facilitates formula simplification, normalization, and modular proof development by allowing local rewrites that maintain logical properties and enabling the transfer of lemmas across contexts where extensionality holds.

Reversal

The reversal is substituting non-equivalent formulas or performing rewrites in contexts where equivalence does not guarantee interchangeability, which can alter truth values and derivability.

Boundary

Applicable in extensional logical contexts and standard propositional/predicate calculi; it fails in intensional logics, many modal contexts, and in places where variable capture or scope changes would occur, unless additional justification is given.

Semantic Tension

There is tension between syntactic replacement (pure symbol manipulation) and semantic equivalence: two formulas may be equivalent extensionally yet not interchangeable within intensional contexts, producing a subtle competition between formal equivalence and contextual meaning.

Synthesis

Substitution of Equivalents: a core rewriting rule in extensional logics that permits replacing provably equivalent subformulas to simplify or transform formulas, valid only where extensionality and binding constraints ensure interchangeability.