Theorem opreu2reurex

Description: There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023) (Revised by AV, 1-Jul-2023)

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

Proof

Step	Hyp	Ref	Expression
1		opreu2reurex.a	⊢ ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ → ( 𝜑 ↔ 𝜒 ) )
2		eqcom	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ↔ ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
3		vex	⊢ 𝑎 ∈ V
4		vex	⊢ 𝑏 ∈ V
5	3 4	opth	⊢ ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) )
6	2 5	bitri	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ↔ ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) )
7	6	imbi2i	⊢ ( ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ↔ ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) )
8	7	a1i	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ) → ( ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ↔ ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) ) )
9	8	2ralbidva	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) ) )
10	9	2rexbiia	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) )
11	10	anbi2i	⊢ ( ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ) ↔ ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) ) )
12	1	reu3op	⊢ ( ∃! 𝑝 ∈ ( 𝐴 × 𝐵 ) 𝜑 ↔ ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ) )
13		2reu4	⊢ ( ( ∃! 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃! 𝑏 ∈ 𝐵 ∃ 𝑎 ∈ 𝐴 𝜒 ) ↔ ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) ) )
14	11 12 13	3bitr4i	⊢ ( ∃! 𝑝 ∈ ( 𝐴 × 𝐵 ) 𝜑 ↔ ( ∃! 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃! 𝑏 ∈ 𝐵 ∃ 𝑎 ∈ 𝐴 𝜒 ) )