Theorem r19.26-2

Description: Restricted quantifier version of 19.26-2 . Version of r19.26 with two quantifiers. (Contributed by NM, 10-Aug-2004)

	Ref	Expression
Assertion	r19.26-2	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.26	⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∀ 𝑦 ∈ 𝐵 𝜓 ) )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∀ 𝑦 ∈ 𝐵 𝜓 ) )
3		r19.26	⊢ ( ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∀ 𝑦 ∈ 𝐵 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) )