Theorem mosubop

Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995)

	Ref	Expression
Hypothesis	mosubop.1	⊢ ∃* 𝑥 𝜑
Assertion	mosubop	⊢ ∃* 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝐴 = ⟨ 𝑦 , 𝑧 ⟩ ∧ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		mosubop.1	⊢ ∃* 𝑥 𝜑
2	1	gen2	⊢ ∀ 𝑦 ∀ 𝑧 ∃* 𝑥 𝜑
3		mosubopt	⊢ ( ∀ 𝑦 ∀ 𝑧 ∃* 𝑥 𝜑 → ∃* 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝐴 = ⟨ 𝑦 , 𝑧 ⟩ ∧ 𝜑 ) )
4	2 3	ax-mp	⊢ ∃* 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝐴 = ⟨ 𝑦 , 𝑧 ⟩ ∧ 𝜑 )