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Metamath Proof Explorer

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

Proof

Step	Hyp	Ref	Expression
1		sb8eu.1	$⊢ Ⅎ y φ$
2	1	nfsb	$⊢ Ⅎ y [w/ x] φ$
3	2	sb8eulem	$⊢ \exists! x φ \leftrightarrow \exists! y [y/ x] φ$