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Metamath Proof Explorer

Theorem exanali

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996) (Proof shortened by Wolf Lammen, 4-Sep-2014)

		Ref	Expression
	Assertion	exanali	$⊢ \exists x (φ \land \neg ψ) \leftrightarrow \neg \forall x (φ \to ψ)$

Proof

Step	Hyp	Ref	Expression
1		annim	$⊢ (φ \land \neg ψ) \leftrightarrow \neg (φ \to ψ)$
2	1	exbii	$⊢ \exists x (φ \land \neg ψ) \leftrightarrow \exists x \neg (φ \to ψ)$
3		exnal	$⊢ \exists x \neg (φ \to ψ) \leftrightarrow \neg \forall x (φ \to ψ)$
4	2 3	bitri	$⊢ \exists x (φ \land \neg ψ) \leftrightarrow \neg \forall x (φ \to ψ)$