Theorem nfr 1761

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 1545	. 2 ⊢ (Ⅎxφ ↔ ∀x(φ → ∀xφ))
2		sp 1747	. 2 ⊢ (∀x(φ → ∀xφ) → (φ → ∀xφ))
3	1, 2	sylbi 187	1 ⊢ (Ⅎxφ → (φ → ∀xφ))

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  nfri  1762  nfrd  1763  19.21t  1795  19.23t  1800  nfimd  1808  nfaldOLD  1853  spimt  1974  sbft  2025