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Metamath Proof Explorer

Theorem euor

Description: Introduce a disjunct into a unique existential quantifier. For a version requiring disjoint variables, but fewer axioms, see euorv . (Contributed by NM, 21-Oct-2005)

		Ref	Expression
	Hypothesis	euor.nf	$⊢ Ⅎ x φ$
	Assertion	euor	$⊢ (\neg φ \land \exists! x ψ) \to \exists! x (φ \lor ψ)$

Proof

Step	Hyp	Ref	Expression
1		euor.nf	$⊢ Ⅎ x φ$
2	1	nfn	$⊢ Ⅎ x \neg φ$
3		biorf	$⊢ \neg φ \to (ψ \leftrightarrow (φ \lor ψ))$
4	2 3	eubid	$⊢ \neg φ \to (\exists! x ψ \leftrightarrow \exists! x (φ \lor ψ))$
5	4	biimpa	$⊢ (\neg φ \land \exists! x ψ) \to \exists! x (φ \lor ψ)$