Theorem aiotaexaiotaiota

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

	Ref	Expression
Assertion	aiotaexaiotaiota	⊢ ( ( ℩' 𝑥 𝜑 ) ∈ V ↔ ( ℩ 𝑥 𝜑 ) = ( ℩' 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		aiotaexb	⊢ ( ∃! 𝑥 𝜑 ↔ ( ℩' 𝑥 𝜑 ) ∈ V )
2		reuaiotaiota	⊢ ( ∃! 𝑥 𝜑 ↔ ( ℩ 𝑥 𝜑 ) = ( ℩' 𝑥 𝜑 ) )
3	1 2	bitr3i	⊢ ( ( ℩' 𝑥 𝜑 ) ∈ V ↔ ( ℩ 𝑥 𝜑 ) = ( ℩' 𝑥 𝜑 ) )