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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	\|- F/ y ph
	Assertion	sb8eu	\|- ( E! x ph <-> E! y [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		sb8eu.1	\|- F/ y ph
2	1	nfsb	\|- F/ y [ w / x ] ph
3	2	sb8eulem	\|- ( E! x ph <-> E! y [ y / x ] ph )