Theorem hbth 1836

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

This and later "hypothesis-building" lemmas, with labels starting "hb...", allow 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 1834), but keeps its interest in some cases. (Revised by BJ, 23-Sep-2022.)

Hypothesis

Ref	Expression
hbth.1	⊢ 𝜑

Assertion

Ref	Expression
hbth	⊢ (𝜑 → ∀𝑥𝜑)

Proof of Theorem hbth

Step	Hyp	Ref	Expression
1		hbth.1	. . 3 ⊢ 𝜑
2	1	ax-gen 1828	. 2 ⊢ ∀𝑥𝜑
3	2	a1i 11	1 ⊢ (𝜑 → ∀𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-gen 1828

This theorem is used by:  speiv  2005  spfalw  2013