Theorem 2ralbiim 3132

Description: Split a biconditional and distribute two restricted universal quantifiers, analogous to 2albiim 1890 and ralbiim 3113. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Assertion

Ref	Expression
2ralbiim	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑)))

Proof of Theorem 2ralbiim

Step	Hyp	Ref	Expression
1		ralbiim 3113	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ↔ 𝜓) ↔ (∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑)))
2	1	ralbii 3093	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ↔ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑)))
3		r19.26 3111	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑)))
4	2, 3	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3062

This theorem is referenced by:  thincciso  49102