This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sb8ef

Description: Substitution of variable in existential quantifier. Version of sb8e with a disjoint variable condition, not requiring ax-13 . (Contributed by NM, 12-Aug-1993) (Revised by Wolf Lammen, 19-Jan-2023)

		Ref	Expression
	Hypothesis	sb8f.nf	$⊢ Ⅎ y φ$
	Assertion	sb8ef	$⊢ \exists x φ \leftrightarrow \exists y [y/ x] φ$

Proof

Step	Hyp	Ref	Expression
1		sb8f.nf	$⊢ Ⅎ y φ$
2		nfs1v	$⊢ Ⅎ x [y/ x] φ$
3		sbequ12	$⊢ x = y \to (φ \leftrightarrow [y/ x] φ)$
4	1 2 3	cbvexv1	$⊢ \exists x φ \leftrightarrow \exists y [y/ x] φ$