Theorem r2exf

Description: Double restricted existential quantification. For a version based on fewer axioms see r2ex . (Contributed by Mario Carneiro, 14-Oct-2016) Use r2exlem . (Revised by Wolf Lammen, 10-Jan-2020)

	Ref	Expression
Hypothesis	r2exf.1	⊢ Ⅎ 𝑦 𝐴
Assertion	r2exf	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		r2exf.1	⊢ Ⅎ 𝑦 𝐴
2	1	r2alf	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ¬ 𝜑 ) )
3	2	r2exlem	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) )