Theorem nfr 1566

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 1509	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		sp 1559	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
3	1, 2	sylbi 121	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1395  Ⅎwnf 1508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-4 1558

This theorem depends on definitions:  df-bi 117  df-nf 1509

This theorem is referenced by:  nfri  1567  nfrd  1568  nfimd  1633  19.23t  1725  equs5or  1878  sbequi  1887  sbft  1896  sbcomxyyz  2025  rgen2a  2586