Theorem nfnf

Description: If x is not free in ph , then it is not free in F/ y ph . (Contributed by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 30-Dec-2017)

	Ref	Expression
Hypothesis	nfnf.1	⊢ Ⅎ 𝑥 𝜑
Assertion	nfnf	⊢ Ⅎ 𝑥 Ⅎ 𝑦 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfnf.1	⊢ Ⅎ 𝑥 𝜑
2		df-nf	⊢ ( Ⅎ 𝑦 𝜑 ↔ ( ∃ 𝑦 𝜑 → ∀ 𝑦 𝜑 ) )
3	1	nfex	⊢ Ⅎ 𝑥 ∃ 𝑦 𝜑
4	1	nfal	⊢ Ⅎ 𝑥 ∀ 𝑦 𝜑
5	3 4	nfim	⊢ Ⅎ 𝑥 ( ∃ 𝑦 𝜑 → ∀ 𝑦 𝜑 )
6	2 5	nfxfr	⊢ Ⅎ 𝑥 Ⅎ 𝑦 𝜑