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Metamath Proof Explorer

Theorem 19.9v

Description: Version of 19.9 with a disjoint variable condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v . (Contributed by NM, 28-May-1995) Remove dependency on ax-7 . (Revised by Wolf Lammen, 4-Dec-2017)

		Ref	Expression
	Assertion	19.9v	$⊢ \exists x φ \leftrightarrow φ$

Proof

Step	Hyp	Ref	Expression
1		ax5e	$⊢ \exists x φ \to φ$
2		19.8v	$⊢ φ \to \exists x φ$
3	1 2	impbii	$⊢ \exists x φ \leftrightarrow φ$