Theorem dfiota4

Description: The iota operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017) (Proof shortened by JJ, 28-Oct-2021)

	Ref	Expression
Assertion	dfiota4	⊢ ( ℩ 𝑥 𝜑 ) = if ( ∃! 𝑥 𝜑 , ∪ { 𝑥 ∣ 𝜑 } , ∅ )

Proof

Step	Hyp	Ref	Expression
1		iotauni	⊢ ( ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) = ∪ { 𝑥 ∣ 𝜑 } )
2		iotanul	⊢ ( ¬ ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) = ∅ )
3		ifval	⊢ ( ( ℩ 𝑥 𝜑 ) = if ( ∃! 𝑥 𝜑 , ∪ { 𝑥 ∣ 𝜑 } , ∅ ) ↔ ( ( ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) = ∪ { 𝑥 ∣ 𝜑 } ) ∧ ( ¬ ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) = ∅ ) ) )
4	1 2 3	mpbir2an	⊢ ( ℩ 𝑥 𝜑 ) = if ( ∃! 𝑥 𝜑 , ∪ { 𝑥 ∣ 𝜑 } , ∅ )