2moswapv

Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 10-Apr-2004) (Revised by GG, 22-Aug-2023) Factor out common proof lines with moexexvw . (Revised by Wolf Lammen, 2-Oct-2023)

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

Proof

Step	Hyp	Ref	Expression
1		nfe1	$⊢ Ⅎ y \exists y φ$
2	1	nfmov	$⊢ Ⅎ y \exists^{*} x \exists y φ$
3		nfe1	$⊢ Ⅎ x \exists x (\exists y φ \land φ)$
4	3	nfmov	$⊢ Ⅎ x \exists^{*} y \exists x (\exists y φ \land φ)$
5	1 2 4	moexexlem	$⊢ (\exists^{} x \exists y φ \land \forall x \exists^{} y φ) \to \exists^{*} y \exists x (\exists y φ \land φ)$
6	5	expcom	$⊢ \forall x \exists^{} y φ \to (\exists^{} x \exists y φ \to \exists^{*} y \exists x (\exists y φ \land φ))$
7		19.8a	$⊢ φ \to \exists y φ$
8	7	pm4.71ri	$⊢ φ \leftrightarrow (\exists y φ \land φ)$
9	8	exbii	$⊢ \exists x φ \leftrightarrow \exists x (\exists y φ \land φ)$
10	9	mobii	$⊢ \exists^{} y \exists x φ \leftrightarrow \exists^{} y \exists x (\exists y φ \land φ)$
11	6 10	imbitrrdi	$⊢ \forall x \exists^{} y φ \to (\exists^{} x \exists y φ \to \exists^{*} y \exists x φ)$

Metamath Proof Explorer

Theorem 2moswapv

Proof