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Metamath Proof Explorer

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 )