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Metamath Proof Explorer

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 )