Theorem albiim

Description: Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993)

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

Proof

Step	Hyp	Ref	Expression
1		dfbi2	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) )
2	1	albii	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ↔ ∀ 𝑥 ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) )
3		19.26	⊢ ( ∀ 𝑥 ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜓 → 𝜑 ) ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜓 → 𝜑 ) ) )