Theorem nfrd 1518

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 1516	. 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1351  Ⅎwnf 1458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-4 1508

This theorem depends on definitions:  df-bi 117  df-nf 1459

This theorem is referenced by:  nfan1  1562  nfim1  1569  alrimdd  1607  spimed  1738  cbv2  1747  nfald  1758  sbied  1786  cbvexd  1925  sbcomxyyz  1970  hbsbd  1980  dvelimALT  2008  dvelimfv  2009  hbeud  2046  abidnf  2903  eusvnfb  4448