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Metamath Proof Explorer

Theorem moexex

Description: "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 . Use the version moexexvw when possible. (Contributed by NM, 3-Dec-2001) (Proof shortened by Wolf Lammen, 28-Dec-2018) Factor out common proof lines with moexexvw . (Revised by Wolf Lammen, 2-Oct-2023) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	moexex.1	$⊢ Ⅎ y φ$
	Assertion	moexex	$⊢ (\exists^{} x φ \land \forall x \exists^{} y ψ) \to \exists^{*} y \exists x (φ \land ψ)$

Proof

Step	Hyp	Ref	Expression
1		moexex.1	$⊢ Ⅎ y φ$
2	1	nfmo	$⊢ Ⅎ y \exists^{*} x φ$
3		nfe1	$⊢ Ⅎ x \exists x (φ \land ψ)$
4	3	nfmo	$⊢ Ⅎ x \exists^{*} y \exists x (φ \land ψ)$
5	1 2 4	moexexlem	$⊢ (\exists^{} x φ \land \forall x \exists^{} y ψ) \to \exists^{*} y \exists x (φ \land ψ)$