Theorem nfs1

Description: If y is not free in ph , x is not free in [ y / x ] ph . Usage of this theorem is discouraged because it depends on ax-13 . Check out nfs1v for a version requiring less axioms. (Contributed by Mario Carneiro, 11-Aug-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	nfs1.1	⊢ Ⅎ 𝑦 𝜑
Assertion	nfs1	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfs1.1	⊢ Ⅎ 𝑦 𝜑
2	1	nf5ri	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3	2	hbsb3	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
4	3	nf5i	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑