Theorem nfrd 1573

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 1571	. 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1400  Ⅎwnf 1513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-4 1563

This theorem depends on definitions:  df-bi 117  df-nf 1514

This theorem is referenced by:  nfan1  1617  nfim1  1624  alrimdd  1662  spimed  1793  cbv2  1802  nfald  1813  sbied  1841  cbvexd  1983  sbcomxyyz  2032  hbsbd  2042  dvelimALT  2070  dvelimfv  2071  hbeud  2108  abidnf  2994  eusvnfb  4595