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Theorem hbth 1805
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 (𝜑 → ∀𝑥𝜑) from smaller formulas of this form. These are useful for constructing hypotheses that state "𝑥 is (effectively) not free in 𝜑". (Contributed by NM, 11-May-1993.) This hb* idiom is generally being replaced by the nf* idiom (see nfth 1803), but keeps its interest in some cases. (Revised by BJ, 23-Sep-2022.)

Ref Expression
hbth.1 𝜑
Ref Expression
hbth (𝜑 → ∀𝑥𝜑)

Proof of Theorem hbth
StepHypRef Expression
1 hbth.1 . . 3 𝜑
21ax-gen 1797 . 2 𝑥𝜑
32a1i 11 1 (𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-gen 1797
This theorem is referenced by:  speiv  1976  spfalw  2004
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