Theorem rmorabex

Description: Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017)

	Ref	Expression
Assertion	rmorabex	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V )

Proof

Step	Hyp	Ref	Expression
1		moabex	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∈ V )
2		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
3		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
4	3	eleq1i	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V ↔ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∈ V )
5	1 2 4	3imtr4i	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V )