Theorem 2albiim

Description: Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005)

	Ref	Expression
Assertion	2albiim	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ∀ 𝑦 ( 𝜓 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		albiim	⊢ ( ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) ↔ ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑦 ( 𝜓 → 𝜑 ) ) )
2	1	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) ↔ ∀ 𝑥 ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑦 ( 𝜓 → 𝜑 ) ) )
3		19.26	⊢ ( ∀ 𝑥 ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑦 ( 𝜓 → 𝜑 ) ) ↔ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ∀ 𝑦 ( 𝜓 → 𝜑 ) ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ∀ 𝑦 ( 𝜓 → 𝜑 ) ) )