Theorem nfrd 1569

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 1567	. 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1396  Ⅎwnf 1509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-4 1559

This theorem depends on definitions:  df-bi 117  df-nf 1510

This theorem is referenced by:  nfan1  1613  nfim1  1620  alrimdd  1658  spimed  1788  cbv2  1797  nfald  1808  sbied  1836  cbvexd  1976  sbcomxyyz  2025  hbsbd  2035  dvelimALT  2063  dvelimfv  2064  hbeud  2101  abidnf  2975  eusvnfb  4557