Theorem r2alan

Description: Double restricted universal quantification, special case. (Contributed by Peter Mazsa, 17-Jun-2020)

	Ref	Expression
Assertion	r2alan	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) → 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		impexp	⊢ ( ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) → 𝜓 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝜑 → 𝜓 ) ) )
2	1	2albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) → 𝜓 ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝜑 → 𝜓 ) ) )
3		r2al	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝜑 → 𝜓 ) ) )
4	2 3	bitr4i	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) → 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 → 𝜓 ) )