Theorem nfdv 1865

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

Hypothesis

Ref	Expression
nfdv.1	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
nfdv	⊢ (𝜑 → Ⅎ𝑥𝜓)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem nfdv

Step	Hyp	Ref	Expression
1		nfdv.1	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
2	1	alrimiv 1862	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
3		df-nf 1449	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
4	2, 3	sylibr 133	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

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-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-17 1514

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

This theorem is referenced by: (None)