Theorem nfsb4t

Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (closed form of nfsb4 ). Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 7-Apr-2004) (Revised by Mario Carneiro, 4-Oct-2016) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	nfsb4t	⊢ ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sbequ12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
2	1	sps	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
3	2	drnf2	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑧 𝜑 ↔ Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
4	3	biimpd	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑧 𝜑 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
5	4	spsd	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
6	5	impcom	⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )
7	6	a1d	⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
8		nfnf1	⊢ Ⅎ 𝑧 Ⅎ 𝑧 𝜑
9	8	nfal	⊢ Ⅎ 𝑧 ∀ 𝑥 Ⅎ 𝑧 𝜑
10		nfnae	⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦
11	9 10	nfan	⊢ Ⅎ 𝑧 ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 )
12		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 Ⅎ 𝑧 𝜑
13		nfnae	⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦
14	12 13	nfan	⊢ Ⅎ 𝑥 ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 )
15		sp	⊢ ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → Ⅎ 𝑧 𝜑 )
16	15	adantr	⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑧 𝜑 )
17		nfsb2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
18	17	adantl	⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
19	1	a1i	⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) ) )
20	11 14 16 18 19	dvelimdf	⊢ ( ( ∀ 𝑥 Ⅎ 𝑧 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
21	7 20	pm2.61dan	⊢ ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )