Theorem reu6i

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Assertion	reu6i	⊢ ( ( 𝐵 ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝑥 = 𝐵 ) ) → ∃! 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		eqeq2	⊢ ( 𝑦 = 𝐵 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝐵 ) )
2	1	bibi2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝜑 ↔ 𝑥 = 𝑦 ) ↔ ( 𝜑 ↔ 𝑥 = 𝐵 ) ) )
3	2	ralbidv	⊢ ( 𝑦 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝑥 = 𝐵 ) ) )
4	3	rspcev	⊢ ( ( 𝐵 ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝑥 = 𝐵 ) ) → ∃ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝑥 = 𝑦 ) )
5		reu6	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝑥 = 𝑦 ) )
6	4 5	sylibr	⊢ ( ( 𝐵 ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝑥 = 𝐵 ) ) → ∃! 𝑥 ∈ 𝐴 𝜑 )