Theorem iotaexeu

Description: The iota class exists. This theorem does not require ax-nul for its proof. (Contributed by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Assertion	iotaexeu	⊢ ( ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		iotaval	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ( ℩ 𝑥 𝜑 ) = 𝑦 )
2	1	eqcomd	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → 𝑦 = ( ℩ 𝑥 𝜑 ) )
3	2	eximi	⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∃ 𝑦 𝑦 = ( ℩ 𝑥 𝜑 ) )
4		eu6	⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) )
5		isset	⊢ ( ( ℩ 𝑥 𝜑 ) ∈ V ↔ ∃ 𝑦 𝑦 = ( ℩ 𝑥 𝜑 ) )
6	3 4 5	3imtr4i	⊢ ( ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) ∈ V )