Theorem hbth 697

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 677	. 2 ⊢ ∀xφ
3	2	a1i 7	1 ⊢ (φ → ∀xφ)

Syntax hints:   → wi 2  ∀wal 672

This theorem is referenced by:  sbie 904  bieu 1014  bimo 1031  biral 1223  birex 1224

This theorem was proved from axioms:  ax-1 3  ax-mp 6  ax-gen 677

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