Metamath Proof Explorer

Theorem wl-dfrexex

Description: Alternate definition of the restricted existential quantification ( df-wl-rex ), according to the pattern given in df-wl-ral . (Contributed by Wolf Lammen, 25-May-2023)

		Ref	Expression
	Assertion	wl-dfrexex	$⊢ \exists x : A φ \leftrightarrow \exists y (y \in A \land \exists x (x = y \land φ))$

Proof

Step	Hyp	Ref	Expression
1		df-wl-ral	$⊢ \forall x : A \neg φ \leftrightarrow \forall y (y \in A \to \forall x (x = y \to \neg φ))$
2	1	notbii	$⊢ \neg \forall x : A \neg φ \leftrightarrow \neg \forall y (y \in A \to \forall x (x = y \to \neg φ))$
3		df-wl-rex	$⊢ \exists x : A φ \leftrightarrow \neg \forall x : A \neg φ$
4		alinexa	$⊢ \forall x (x = y \to \neg φ) \leftrightarrow \neg \exists x (x = y \land φ)$
5	4	imbi2i	$⊢ (y \in A \to \forall x (x = y \to \neg φ)) \leftrightarrow (y \in A \to \neg \exists x (x = y \land φ))$
6	5	albii	$⊢ \forall y (y \in A \to \forall x (x = y \to \neg φ)) \leftrightarrow \forall y (y \in A \to \neg \exists x (x = y \land φ))$
7		alinexa	$⊢ \forall y (y \in A \to \neg \exists x (x = y \land φ)) \leftrightarrow \neg \exists y (y \in A \land \exists x (x = y \land φ))$
8	6 7	bitri	$⊢ \forall y (y \in A \to \forall x (x = y \to \neg φ)) \leftrightarrow \neg \exists y (y \in A \land \exists x (x = y \land φ))$
9	8	con2bii	$⊢ \exists y (y \in A \land \exists x (x = y \land φ)) \leftrightarrow \neg \forall y (y \in A \to \forall x (x = y \to \neg φ))$
10	2 3 9	3bitr4i	$⊢ \exists x : A φ \leftrightarrow \exists y (y \in A \land \exists x (x = y \land φ))$