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	⊢ ( ( ¬ 𝜑 ∧ ∃! 𝑥 𝜓 ) → ∃! 𝑥 ( 𝜑 ∨ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		euor.nf	⊢ Ⅎ 𝑥 𝜑
2	1	nfn	⊢ Ⅎ 𝑥 ¬ 𝜑
3		biorf	⊢ ( ¬ 𝜑 → ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) )
4	2 3	eubid	⊢ ( ¬ 𝜑 → ( ∃! 𝑥 𝜓 ↔ ∃! 𝑥 ( 𝜑 ∨ 𝜓 ) ) )
5	4	biimpa	⊢ ( ( ¬ 𝜑 ∧ ∃! 𝑥 𝜓 ) → ∃! 𝑥 ( 𝜑 ∨ 𝜓 ) )