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

Metamath Proof Explorer

Theorem rexanali

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005) (Proof shortened by Wolf Lammen, 27-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		dfrex2	$⊢ \exists x \in A (φ \land \neg ψ) \leftrightarrow \neg \forall x \in A \neg (φ \land \neg ψ)$
2		iman	$⊢ (φ \to ψ) \leftrightarrow \neg (φ \land \neg ψ)$
3	2	ralbii	$⊢ \forall x \in A (φ \to ψ) \leftrightarrow \forall x \in A \neg (φ \land \neg ψ)$
4	1 3	xchbinxr	$⊢ \exists x \in A (φ \land \neg ψ) \leftrightarrow \neg \forall x \in A (φ \to ψ)$