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

Metamath Proof Explorer

Theorem nrexdv

Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003) (Proof shortened by Wolf Lammen, 5-Jan-2020)

		Ref	Expression
	Hypothesis	nrexdv.1	$⊢ (φ \land x \in A) \to \neg ψ$
	Assertion	nrexdv	$⊢ φ \to \neg \exists x \in A ψ$

Proof

Step	Hyp	Ref	Expression
1		nrexdv.1	$⊢ (φ \land x \in A) \to \neg ψ$
2	1	ralrimiva	$⊢ φ \to \forall x \in A \neg ψ$
3		ralnex	$⊢ \forall x \in A \neg ψ \leftrightarrow \neg \exists x \in A ψ$
4	2 3	sylib	$⊢ φ \to \neg \exists x \in A ψ$