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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	$⊢ \exists! x φ \leftrightarrow \exists! y [y/ x] φ$

Proof

Step	Hyp	Ref	Expression
1		sb8euv.nf	$⊢ Ⅎ y φ$
2	1	nfsbv	$⊢ Ⅎ y [w/ x] φ$
3	2	sb8eulem	$⊢ \exists! x φ \leftrightarrow \exists! y [y/ x] φ$