2euswapv

Description: A condition allowing to swap an existential quantifier and a unique existential quantifier. Version of 2euswap with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 10-Apr-2004) (Revised by GG, 22-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		excomim	$⊢ \exists x \exists y φ \to \exists y \exists x φ$
2	1	a1i	$⊢ \forall x \exists^{*} y φ \to (\exists x \exists y φ \to \exists y \exists x φ)$
3		2moswapv	$⊢ \forall x \exists^{} y φ \to (\exists^{} x \exists y φ \to \exists^{*} y \exists x φ)$
4	2 3	anim12d	$⊢ \forall x \exists^{} y φ \to ((\exists x \exists y φ \land \exists^{} x \exists y φ) \to (\exists y \exists x φ \land \exists^{*} y \exists x φ))$
5		df-eu	$⊢ \exists! x \exists y φ \leftrightarrow (\exists x \exists y φ \land \exists^{*} x \exists y φ)$
6		df-eu	$⊢ \exists! y \exists x φ \leftrightarrow (\exists y \exists x φ \land \exists^{*} y \exists x φ)$
7	4 5 6	3imtr4g	$⊢ \forall x \exists^{*} y φ \to (\exists! x \exists y φ \to \exists! y \exists x φ)$

Metamath Proof Explorer

Theorem 2euswapv

Proof