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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	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑

Proof

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