Theorem reu6dv

Description: A condition which implies existential uniqueness. (Contributed by Thierry Arnoux, 13-Oct-2025)

	Ref	Expression
Hypotheses	reu6d.1	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
	reu6d.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝑥 = 𝐵 ) )
Assertion	reu6dv	⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		reu6d.1	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
2		reu6d.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝑥 = 𝐵 ) )
3	2	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝑥 = 𝐵 ) )
4		reu6i	⊢ ( ( 𝐵 ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝑥 = 𝐵 ) ) → ∃! 𝑥 ∈ 𝐴 𝜓 )
5	1 3 4	syl2anc	⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜓 )