Theorem hbth 1552

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, 5-Aug-1993.)

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 1546	. 2 ⊢ ∀xφ
3	2	a1i 10	1 ⊢ (φ → ∀xφ)

This proof depends on syntax axioms:   → wi 4  ∀wal 1540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-gen 1546

This theorem is used by:  nfth  1553  spfalw  1672  spimehOLD  1821