Theorem nf5dv 2144

Description: Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1787 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 1913	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		nf5dv.1	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3	1, 2	nf5dh 2143	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1537  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-10 2137

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787

This theorem is referenced by: (None)