Theorem nfrd 1514

Description: Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfrd.1	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfrd	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Proof of Theorem nfrd

Step	Hyp	Ref	Expression
1		nfrd.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜓)
2		nfr 1512	. 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1347  Ⅎwnf 1454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-4 1504

This theorem depends on definitions:  df-bi 116  df-nf 1455

This theorem is referenced by:  nfan1  1558  nfim1  1565  alrimdd  1603  spimed  1734  cbv2  1743  nfald  1754  sbied  1782  cbvexd  1921  sbcomxyyz  1966  hbsbd  1976  dvelimALT  2004  dvelimfv  2005  hbeud  2042  abidnf  2899  eusvnfb  4440