Theorem hbth

Description: No variable is (effectively) free in a theorem.

This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form |- ( ph -> A. x ph ) from smaller formulas of this form. These are useful for constructing hypotheses that state " x is (effectively) not free in ph ". (Contributed by NM, 11-May-1993) This hb* idiom is generally being replaced by the nf* idiom (see nfth ), but keeps its interest in some cases. (Revised by BJ, 23-Sep-2022)

	Ref	Expression
Hypothesis	hbth.1	$$⊢\phi$$
Assertion	hbth	$$⊢\phi \to \forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		hbth.1	$$⊢\phi$$
2	1	ax-gen	$$⊢\forall x\phi$$
3	2	a1i	$$⊢\phi \to \forall x\phi$$