Metamath Proof Explorer


Theorem nfs1v

Description: The setvar x is not free in [ y / x ] ph when x and y are distinct. (Contributed by Mario Carneiro, 11-Aug-2016) Shorten nfs1v and hbs1 combined. (Revised by Wolf Lammen, 28-Jul-2022)

Ref Expression
Assertion nfs1v 𝑥 [ 𝑦 / 𝑥 ] 𝜑

Proof

Step Hyp Ref Expression
1 sb6 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) )
2 nfa1 𝑥𝑥 ( 𝑥 = 𝑦𝜑 )
3 1 2 nfxfr 𝑥 [ 𝑦 / 𝑥 ] 𝜑