Theorem nf5dv 2025

Description: Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.)

Hypothesis

Ref	Expression
nf5dv.1	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
nf5dv	⊢ (𝜑 → Ⅎ𝑥𝜓)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem nf5dv

Step	Hyp	Ref	Expression
1		nf5dv.1	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
2	1	alrimiv 1855	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
3		nf5-1 2023	. 2 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → Ⅎ𝑥𝜓)
4	2, 3	syl 17	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-10 2019

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)