Theorem nfr 1452

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)

Assertion

Ref	Expression
nfr	⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem nfr

Step	Hyp	Ref	Expression
1		df-nf 1391	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		sp 1442	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
3	1, 2	sylbi 119	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1283  Ⅎwnf 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-4 1441

This theorem depends on definitions:  df-bi 115  df-nf 1391

This theorem is referenced by:  nfri  1453  nfrd  1454  nfimd  1518  19.23t  1608  equs5or  1752  sbequi  1761  sbft  1770  sbcomxyyz  1888  rgen2a  2418