Theorem sb8euv

Description: Variable substitution in unique existential quantifier. Version of sb8eu requiring more disjoint variables, but fewer axioms. (Contributed by NM, 7-Aug-1994) (Revised by Wolf Lammen, 7-Feb-2023)

	Ref	Expression
Hypothesis	sb8euv.nf	⊢ Ⅎ 𝑦 𝜑
Assertion	sb8euv	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb8euv.nf	⊢ Ⅎ 𝑦 𝜑
2	1	nfsbv	⊢ Ⅎ 𝑦 [ 𝑤 / 𝑥 ] 𝜑
3	2	sb8eulem	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )