Theorem nfr 1542

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 1485	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		sp 1535	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
3	1, 2	sylbi 121	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1371  Ⅎwnf 1484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-4 1534

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  nfri  1543  nfrd  1544  nfimd  1609  19.23t  1701  equs5or  1854  sbequi  1863  sbft  1872  sbcomxyyz  2001  rgen2a  2562