Alphabetic Substitution is Semantically Equivalent/Existential Quantifier

Theorem

Let $\map \phi x$ be a WFF of predicate logic.

Let $z$ be free for $x$ in $\phi$.

Let $z$ not occur freely in $\phi$.

Let $\map \phi z$ be the result of the alphabetic substitution of $z$ for $x$.

Then:

$\exists x: \map \phi x$ and $\exists z: \map \phi z$ are semantically equivalent

Proof

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Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Exercise $\text{II}.8.25$