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	$$⊢Ⅎy\phi$$
Assertion	sb8eu	$$⊢\exists !x\phi \leftrightarrow \exists !y\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sb8eu.1	$$⊢Ⅎy\phi$$
2	1	nfsb	$$⊢Ⅎy\left[w/x\right]\phi$$
3	2	sb8eulem	$$⊢\exists !x\phi \leftrightarrow \exists !y\left[y/x\right]\phi$$