Theorem nfsbv

Description: If z is not free in ph , it is not free in [ y / x ] ph when z is distinct from x and y . Version of nfsb requiring more disjoint variables, but fewer axioms. (Contributed by Mario Carneiro, 11-Aug-2016) (Revised by Wolf Lammen, 7-Feb-2023) Remove disjoint variable condition on x , y . (Revised by Steven Nguyen, 13-Aug-2023)

	Ref	Expression
Hypothesis	nfsbv.nf	⊢ Ⅎ 𝑧 𝜑
Assertion	nfsbv	⊢ Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfsbv.nf	⊢ Ⅎ 𝑧 𝜑
2		df-sb	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑤 ( 𝑤 = 𝑦 → ∀ 𝑥 ( 𝑥 = 𝑤 → 𝜑 ) ) )
3		nfv	⊢ Ⅎ 𝑧 𝑤 = 𝑦
4		nfv	⊢ Ⅎ 𝑧 𝑥 = 𝑤
5	4 1	nfim	⊢ Ⅎ 𝑧 ( 𝑥 = 𝑤 → 𝜑 )
6	5	nfal	⊢ Ⅎ 𝑧 ∀ 𝑥 ( 𝑥 = 𝑤 → 𝜑 )
7	3 6	nfim	⊢ Ⅎ 𝑧 ( 𝑤 = 𝑦 → ∀ 𝑥 ( 𝑥 = 𝑤 → 𝜑 ) )
8	7	nfal	⊢ Ⅎ 𝑧 ∀ 𝑤 ( 𝑤 = 𝑦 → ∀ 𝑥 ( 𝑥 = 𝑤 → 𝜑 ) )
9	2 8	nfxfr	⊢ Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑