Am new to Logic, and am having difficulty with this proof of Proposition 2.10 of Mendelson's Book "Introduction to Logic". The proposition describes a rule termed "Rule C", whereby given a well-formed formula (wff) $\mathcal{B}(x)$, with free variable $x$, Rule C permits passing from $(\exists x)(\mathcal{B}(x))$ to $\mathcal{B}(b)$ where $b$ is a constant (aka Existential Instantiation in other books).
Just to make this post self-contained, I will describe Rule C deductions in detail. Given a set of hypothesis $\Gamma$, a Rule C deduction to $\mathcal{B}$ is denoted as $\Gamma \vdash_C \mathcal{B}$, whereby the deduction involves a sequence of wffs $\mathcal{D}_1,...,\mathcal{D}_n$ with $\mathcal{D}_n=\mathcal{B}$, and where:
For each $i < n$, $\mathcal{D}_i$ is either: (i) an axiom, (ii) a hypothesis in $\Gamma$, (iii) a result of Modus Ponens or Generalization from preceding wffs in the deduction, or (iv) if there is a preceding wf $(\exists x) \mathcal{C}(x)$, and $\mathcal{D}_i=\mathcal{C}(d)$ with $d$ a constant.
$\mathcal{B}$ has no constants introduced from the deduction
Universal Generalization is not made using a variable that is free in some $(\exists x)\mathcal{C}(x)$ to which rule C was applied.
Now, let $\vdash$ represent the usual logical implication. We have the ff.
Proposition 2.10: If $\Gamma \vdash_C \mathcal{B}$, then $\Gamma \vdash \mathcal{B}$
Proof. Let $(\exists y_1)\mathcal{C}_1(y_1),...,(\exists y_k)\mathcal{C}_k(y_k)$ be wffs in order of occurence to which Rule C is applied in the proof of $\Gamma \vdash_C \mathcal{B}$, and let $d_1,...,d_k$ be new constants introduced on each application of Rule C. Then $\Gamma, \mathcal{C}_1(d_1),...,\mathcal{C}_k(d_k) \vdash \mathcal{B}$. Using a basic result of deductions, we also have $\Gamma, \mathcal{C}_1(d_1),..., \mathcal{C}_{k-1}(d_{k-1}) \vdash \mathcal{C}_k(d_k) \rightarrow \mathcal{B}$. Now, replace $d_k$ everywhere by a variable $z$ that does not occur in the proof so that $\Gamma, \mathcal{C}_1(d_1),..., \mathcal{C}_{k-1}(d_{k-1}) \vdash \mathcal{C}_k(z) \rightarrow \mathcal{B}$. Applying Universal Generalization aftewards, we also have $\Gamma, \mathcal{C}_1(d_1),..., \mathcal{C}_{k-1}(d_{k-1}) \vdash (\forall z) (\mathcal{C}_k(z) \rightarrow \mathcal{B})$ (for brevity, I will omit the rest of the proof from this point on...)
What is confusing me is the part in the proof whereby the introduced constant $d_k$ is substituted by a variable $z$, and for which Universal Generalization is subsequently applied to arrive at the term $(\forall z) (\mathcal{C}_k(z) \rightarrow \mathcal{B})$
This seems to make the following deduction possible. For instance, let $\Gamma:=\{\exists x (x < 1)\}$, where $[x < 1]$ is a unary predicate with $1$ as a constant. We have the following:
- $(\exists x)(x < 1)$ - hypothesis
- $d < 1$ - application of Rule C
- $z < 1$ - introduction of new variable $z$ which replaces $d$ (as done in the proof)
- $(\forall z)(z < 1)$ - Universal Generalization (as done in the proof)
This is obviously not correct. What am I missing in my understanding of the proof ?