Theorem 2reucom

Description: Double restricted existential uniqueness commutes. (Contributed by Thierry Arnoux, 4-Jul-2023)

	Ref	Expression
Assertion	2reucom	⊢ ( ∃! 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 𝜑 ↔ ∃! 𝑦 ∈ 𝐵 , 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ancom	⊢ ( ( ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃! 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝜑 ) ↔ ( ∃! 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ) )
2		df-2reu	⊢ ( ∃! 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 𝜑 ↔ ( ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃! 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝜑 ) )
3		df-2reu	⊢ ( ∃! 𝑦 ∈ 𝐵 , 𝑥 ∈ 𝐴 𝜑 ↔ ( ∃! 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ) )
4	1 2 3	3bitr4i	⊢ ( ∃! 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 𝜑 ↔ ∃! 𝑦 ∈ 𝐵 , 𝑥 ∈ 𝐴 𝜑 )