Theorem riota1a

Description: Property of iota. (Contributed by NM, 23-Aug-2011)

	Ref	Expression
Assertion	riota1a	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∃! 𝑥 ∈ 𝐴 𝜑 ) → ( 𝜑 ↔ ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) = 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		ibar	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
2		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
3		iota1	⊢ ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) = 𝑥 ) )
4	2 3	sylbi	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) = 𝑥 ) )
5	1 4	sylan9bb	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∃! 𝑥 ∈ 𝐴 𝜑 ) → ( 𝜑 ↔ ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) = 𝑥 ) )