Theorem nfrd 1508

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 1506	. 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1341  Ⅎwnf 1448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  nfan1  1552  nfim1  1559  alrimdd  1597  spimed  1728  cbv2  1737  nfald  1748  sbied  1776  cbvexd  1915  sbcomxyyz  1960  hbsbd  1970  dvelimALT  1998  dvelimfv  1999  hbeud  2036  abidnf  2894  eusvnfb  4432