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Metamath Proof Explorer

Theorem sb8e

Description: Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring disjoint variables, but fewer axioms, see sb8ef . (Contributed by NM, 12-Aug-1993) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Jim Kingdon, 15-Jan-2018) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sb8.1	$⊢ Ⅎ y φ$
	Assertion	sb8e	$⊢ \exists x φ \leftrightarrow \exists y [y/ x] φ$

Proof

Step	Hyp	Ref	Expression
1		sb8.1	$⊢ Ⅎ y φ$
2	1	nfs1	$⊢ Ⅎ x [y/ x] φ$
3		sbequ12	$⊢ x = y \to (φ \leftrightarrow [y/ x] φ)$
4	1 2 3	cbvex	$⊢ \exists x φ \leftrightarrow \exists y [y/ x] φ$