Theorem opreu2reu

Description: If there is a unique ordered pair fulfilling a wff, then there is a double restricted unique existential qualification fulfilling a corresponding wff. (Contributed by AV, 25-Jun-2023) (Revised by AV, 2-Jul-2023)

	Ref	Expression
Hypothesis	opreu2reurex.a	⊢ ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ → ( 𝜑 ↔ 𝜒 ) )
Assertion	opreu2reu	⊢ ( ∃! 𝑝 ∈ ( 𝐴 × 𝐵 ) 𝜑 → ∃! 𝑎 ∈ 𝐴 ∃! 𝑏 ∈ 𝐵 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		opreu2reurex.a	⊢ ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ → ( 𝜑 ↔ 𝜒 ) )
2	1	opreu2reurex	⊢ ( ∃! 𝑝 ∈ ( 𝐴 × 𝐵 ) 𝜑 ↔ ( ∃! 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃! 𝑏 ∈ 𝐵 ∃ 𝑎 ∈ 𝐴 𝜒 ) )
3		2rexreu	⊢ ( ( ∃! 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃! 𝑏 ∈ 𝐵 ∃ 𝑎 ∈ 𝐴 𝜒 ) → ∃! 𝑎 ∈ 𝐴 ∃! 𝑏 ∈ 𝐵 𝜒 )
4	2 3	sylbi	⊢ ( ∃! 𝑝 ∈ ( 𝐴 × 𝐵 ) 𝜑 → ∃! 𝑎 ∈ 𝐴 ∃! 𝑏 ∈ 𝐵 𝜒 )