2eu1

Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2eu1v when possible. (Contributed by NM, 3-Dec-2001) (Proof shortened by Wolf Lammen, 23-Apr-2023) (New usage is discouraged.)

		Ref	Expression
	Assertion	2eu1	$⊢ \forall x \exists^{*} y φ \to (\exists! x \exists! y φ \leftrightarrow (\exists! x \exists y φ \land \exists! y \exists x φ))$

Proof

Step	Hyp	Ref	Expression
1		2eu2ex	$⊢ \exists! x \exists! y φ \to \exists x \exists y φ$
2		moeu	$⊢ \exists^{*} y φ \leftrightarrow (\exists y φ \to \exists! y φ)$
3	2	albii	$⊢ \forall x \exists^{*} y φ \leftrightarrow \forall x (\exists y φ \to \exists! y φ)$
4		euim	$⊢ (\exists x \exists y φ \land \forall x (\exists y φ \to \exists! y φ)) \to (\exists! x \exists! y φ \to \exists! x \exists y φ)$
5	3 4	sylan2b	$⊢ (\exists x \exists y φ \land \forall x \exists^{*} y φ) \to (\exists! x \exists! y φ \to \exists! x \exists y φ)$
6	5	ex	$⊢ \exists x \exists y φ \to (\forall x \exists^{*} y φ \to (\exists! x \exists! y φ \to \exists! x \exists y φ))$
7	1 6	syl	$⊢ \exists! x \exists! y φ \to (\forall x \exists^{*} y φ \to (\exists! x \exists! y φ \to \exists! x \exists y φ))$
8	7	pm2.43b	$⊢ \forall x \exists^{*} y φ \to (\exists! x \exists! y φ \to \exists! x \exists y φ)$
9		2euswap	$⊢ \forall x \exists^{*} y φ \to (\exists! x \exists y φ \to \exists! y \exists x φ)$
10	8 9	syld	$⊢ \forall x \exists^{*} y φ \to (\exists! x \exists! y φ \to \exists! y \exists x φ)$
11	8 10	jcad	$⊢ \forall x \exists^{*} y φ \to (\exists! x \exists! y φ \to (\exists! x \exists y φ \land \exists! y \exists x φ))$
12		2exeu	$⊢ (\exists! x \exists y φ \land \exists! y \exists x φ) \to \exists! x \exists! y φ$
13	11 12	impbid1	$⊢ \forall x \exists^{*} y φ \to (\exists! x \exists! y φ \leftrightarrow (\exists! x \exists y φ \land \exists! y \exists x φ))$

Metamath Proof Explorer

Theorem 2eu1

Proof