Theorem euorv

Description: Introduce a disjunct into a unique existential quantifier. Version of euor requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023)

	Ref	Expression
Assertion	euorv	⊢ ( ( ¬ 𝜑 ∧ ∃! 𝑥 𝜓 ) → ∃! 𝑥 ( 𝜑 ∨ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		biorf	⊢ ( ¬ 𝜑 → ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) )
2	1	eubidv	⊢ ( ¬ 𝜑 → ( ∃! 𝑥 𝜓 ↔ ∃! 𝑥 ( 𝜑 ∨ 𝜓 ) ) )
3	2	biimpa	⊢ ( ( ¬ 𝜑 ∧ ∃! 𝑥 𝜓 ) → ∃! 𝑥 ( 𝜑 ∨ 𝜓 ) )