Theorem nf5dv 2186

Description: Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1817 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof shortened by Wolf Lammen, 13-Jul-2022.)

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		ax-5 1943	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		nf5dv.1	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3	1, 2	nf5dh 2185	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-10 2179

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by: (None)