Theorem wl-nfs1t

Description: If y is not free in ph , x is not free in [ y / x ] ph . Closed form of nfs1 . (Contributed by Wolf Lammen, 27-Jul-2019)

	Ref	Expression
Assertion	wl-nfs1t	⊢ ( Ⅎ 𝑦 𝜑 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbequ12r	⊢ ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
2	1	equcoms	⊢ ( 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
3	2	sps	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
4	3	drnf1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ↔ Ⅎ 𝑦 𝜑 ) )
5	4	biimprd	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑦 𝜑 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ) )
6		nfsb2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
7	6	a1d	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑦 𝜑 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ) )
8	5 7	pm2.61i	⊢ ( Ⅎ 𝑦 𝜑 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )