Theorem euop2

Description: Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004)

	Ref	Expression
Hypothesis	euop2.1	⊢ 𝐴 ∈ V
Assertion	euop2	⊢ ( ∃! 𝑥 ∃ 𝑦 ( 𝑥 = ⟨ 𝐴 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃! 𝑦 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		euop2.1	⊢ 𝐴 ∈ V
2		opex	⊢ ⟨ 𝐴 , 𝑦 ⟩ ∈ V
3	1	moop2	⊢ ∃* 𝑦 𝑥 = ⟨ 𝐴 , 𝑦 ⟩
4	2 3	euxfr2w	⊢ ( ∃! 𝑥 ∃ 𝑦 ( 𝑥 = ⟨ 𝐴 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃! 𝑦 𝜑 )