Theorem nfsb

Description: If z is not free in ph , then it is not free in [ y / x ] ph when y and z are distinct. See nfsbv for a version with an additional disjoint variable condition on x , z but not requiring ax-13 . (Contributed by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 25-Feb-2024) Usage of this theorem is discouraged because it depends on ax-13 . Use nfsbv instead. (New usage is discouraged.)

	Ref	Expression
Hypothesis	nfsb.1	⊢ Ⅎ 𝑧 𝜑
Assertion	nfsb	⊢ Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfsb.1	⊢ Ⅎ 𝑧 𝜑
2		nftru	⊢ Ⅎ 𝑥 ⊤
3	1	a1i	⊢ ( ⊤ → Ⅎ 𝑧 𝜑 )
4	2 3	nfsbd	⊢ ( ⊤ → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )
5	4	mptru	⊢ Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑