Theorem 2rexbii

Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995)

	Ref	Expression
Hypothesis	2rexbii.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	2rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		2rexbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	rexbii	⊢ ( ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 𝜓 )
3	2	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 )