Equality implies Substitution

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Theorem

Let $\map P x$ denote a Well-Formed Formula which contains $x$ as a free variable.

Then the following are tautologies:

$\forall x: \paren {\map P x \iff \exists y: \paren {y = x \land \map P y} }$

$\forall x: \paren {\map P x \iff \forall y: \paren {y = x \implies \map P y} }$

Note that when $y$ is substituted for $x$ in either formula, it is false in general; compare Confusion of Bound Variables.

Proof

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By Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous:

$\paren {\exists y: y = x \land \forall y: \paren {y = x \implies \map P x} } \implies \exists y: \paren {y = x \land \map P x}$

Then:

\(\ds \)

\(\)

\(\ds x = x\)

Equality is Reflexive

\(\ds \)

\(\leadsto\)

\(\ds \exists y: y = x\)

Existential Generalisation

\(\ds \)

\(\leadsto\)

\(\ds \paren {\forall y: \paren {y = x \implies \map P x} \implies \exists y: \paren {y = x \land \map P x} }\)

Modus Ponendo Ponens

$\Box$

\(\ds \paren { y = x \land \map P y}\)

\(\implies\)

\(\ds \map P x\)

Substitutivity of Equality

\(\text {(1)}: \quad\)

\(\ds \leadsto \ \ \)

\(\ds \exists y: \, \)

\(\ds \paren {y = x \land \map P y}\)

\(\implies\)

\(\ds \map P x\)

Universal Generalisation

\(\text {(2)}: \quad\)

\(\ds \leadsto \ \ \)

\(\ds \forall y: \, \)

\(\ds \paren {y = x \implies \map P y}\)

\(\implies\)

\(\ds \map P x\)

Hypothetical Syllogism with first lemma

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$\Box$

Similarly:

\(\ds \)

\(\)

\(\ds \map P x\)

\(\ds \leadsto \ \ \)

\(\ds y = x\)

\(\implies\)

\(\ds \map P y\)

Substitutivity of Equality

\(\text {(3)}: \quad\)

\(\ds \leadsto \ \ \)

\(\ds \forall y: \, \)

\(\ds y = x\)

\(\implies\)

\(\ds \map P y\)

Universal Generalisation

\(\text {(4)}: \quad\)

\(\ds \leadsto \ \ \)

\(\ds \exists y: \, \)

\(\ds y = x\)

\(\land\)

\(\ds \map P y\)

First lemma

The above two statements comprise the other direction of the biconditional assertions.

Together, $(1)$, $(2)$, $(3)$, and $(4)$ prove the two assertions.

$\blacksquare$

Sources

• 1963: Willard Van Orman Quine: Set Theory and Its Logic: $\S 6.1$
• 1963: Willard Van Orman Quine: Set Theory and Its Logic: $\S 6.2$