Theorem sb8eu

Description: Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring more disjoint variables, but fewer axioms, see sb8euv . (Contributed by NM, 7-Aug-1994) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 24-Aug-2019) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb8eu.1	⊢ Ⅎ 𝑦 𝜑
Assertion	sb8eu	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb8eu.1	⊢ Ⅎ 𝑦 𝜑
2	1	nfsb	⊢ Ⅎ 𝑦 [ 𝑤 / 𝑥 ] 𝜑
3	2	sb8eulem	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )