Theorem 2rexbidva

Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004)

	Ref	Expression
Hypothesis	2ralbidva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	2rexbidva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		2ralbidva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
2	1	anassrs	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	rexbidva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) )
4	3	rexbidva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) )