Theorem ralbiim

Description: Split a biconditional and distribute quantifier. Restricted quantifier version of albiim . (Contributed by NM, 3-Jun-2012)

	Ref	Expression
Assertion	ralbiim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfbi2	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) )
3		r19.26	⊢ ( ∀ 𝑥 ∈ 𝐴 ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜑 ) ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜑 ) ) )