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Metamath Proof Explorer

Theorem 2moex

Description: Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2moexv when possible. (Contributed by NM, 3-Dec-2001) (New usage is discouraged.)

		Ref	Expression
	Assertion	2moex	$⊢ \exists^{} x \exists y φ \to \forall y \exists^{} x φ$

Proof

Step	Hyp	Ref	Expression
1		nfe1	$⊢ Ⅎ y \exists y φ$
2	1	nfmo	$⊢ Ⅎ y \exists^{*} x \exists y φ$
3		19.8a	$⊢ φ \to \exists y φ$
4	3	moimi	$⊢ \exists^{} x \exists y φ \to \exists^{} x φ$
5	2 4	alrimi	$⊢ \exists^{} x \exists y φ \to \forall y \exists^{} x φ$