Theorem reuaiotaiota

Description: The iota and the alternate iota over a wff ph are equal iff there is a unique value x satisfying ph . (Contributed by AV, 25-Aug-2022)

	Ref	Expression
Assertion	reuaiotaiota	⊢ ( ∃! 𝑥 𝜑 ↔ ( ℩ 𝑥 𝜑 ) = ( ℩' 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		euabsneu	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } )
2		reuabaiotaiota	⊢ ( ∃! 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } ↔ ( ℩ 𝑥 𝜑 ) = ( ℩' 𝑥 𝜑 ) )
3	1 2	bitri	⊢ ( ∃! 𝑥 𝜑 ↔ ( ℩ 𝑥 𝜑 ) = ( ℩' 𝑥 𝜑 ) )