Theorem dvelimf

Description: Version of dvelimv without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 1-Oct-2002) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dvelimf.1	⊢ Ⅎ 𝑥 𝜑
	dvelimf.2	⊢ Ⅎ 𝑧 𝜓
	dvelimf.3	⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	dvelimf	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		dvelimf.1	⊢ Ⅎ 𝑥 𝜑
2		dvelimf.2	⊢ Ⅎ 𝑧 𝜓
3		dvelimf.3	⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	2 3	equsal	⊢ ( ∀ 𝑧 ( 𝑧 = 𝑦 → 𝜑 ) ↔ 𝜓 )
5	4	bicomi	⊢ ( 𝜓 ↔ ∀ 𝑧 ( 𝑧 = 𝑦 → 𝜑 ) )
6		nfnae	⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦
7		nfeqf	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝑧 = 𝑦 )
8	7	ancoms	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝑧 = 𝑦 )
9	1	a1i	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝜑 )
10	8 9	nfimd	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 ( 𝑧 = 𝑦 → 𝜑 ) )
11	6 10	nfald2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 ∀ 𝑧 ( 𝑧 = 𝑦 → 𝜑 ) )
12	5 11	nfxfrd	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜓 )