Theorem hbth 1000

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."

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 962	. 2 ⊢ ∀xφ
3	2	a1i 8	1 ⊢ (φ → ∀xφ)

Syntax hints:   → wi 3  ∀wal 953

This theorem is referenced by:  sbie 1195  a12lem1 1375  ralbii 1665  rexbii 1666  sbcralg 1991  sbcrexg 1992  infcvgaux1 7171

This theorem was proved from axioms:  ax-1 4  ax-mp 7  ax-gen 962

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