Axioms of Hilbert Proof System Instance 1 for Predicate Logic are Tautologies

Theorem
Let $\mathscr H$ be Instance 1 of a Hilbert proof system for predicate logic.

Then the axioms of $\mathscr H$ are tautologies.

Axiom 1
This is precisely the statement of Propositional Tautology is Tautology in Predicate Logic.

By Well-Formed Formula is Tautology iff Universal Closure is Tautology, we prove the results for axioms 2 through 11 for the WFFs instead of their universal closures.

Let $\sigma$ be an assignment in a structure $\AA$.

Let $\map { \operatorname{val}_\AA } {\cdot} \sqbrk \sigma$ be the value mapping for $\sigma$ in $\AA$.

Axiom 2: $\mathbf A \implies \paren{ \forall x: \mathbf A }$
We have that $\map { \operatorname{val}_\AA } {\forall x: \mathbf A} \sqbrk \sigma = T$ :


 * $\forall a \in A: \map { \operatorname{val}_\AA } {\mathbf A} \sqbrk { \sigma + \paren{ a / x } } = T$

By assumption, $x$ does not occur freely in $\mathbf A$.

By Value of Formula under Assignment Determined by Free Variables, it follows that:


 * $\forall a \in A: \map { \operatorname{val}_\AA } {\mathbf A} \sqbrk { \sigma + \paren{ a / x } } = \map { \operatorname{val}_\AA } {\mathbf A} \sqbrk { \sigma }$

and hence:


 * $\map { \operatorname{val}_\AA } {\forall x: \mathbf A} \sqbrk \sigma = \map { \operatorname{val}_\AA } {\mathbf A} \sqbrk { \sigma }$

Therefore:

Since $\AA$ and $\sigma$ are arbitrary, it follows that $\mathbf A \implies \paren{ \forall x: \mathbf A }$ is a tautology.