Theorem 2ralbidva

Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 9-Dec-2019)

	Ref	Expression
Hypothesis	2ralbidva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	2ralbidva	⊢ ( 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	ralbidva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝐵 𝜓 ↔ ∀ 𝑦 ∈ 𝐵 𝜒 ) )
4	3	ralbidva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) )