Metamath Proof Explorer


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 φ
Assertion hbth φ x φ

Proof

Step Hyp Ref Expression
1 hbth.1 φ
2 1 ax-gen x φ
3 2 a1i φ x φ