Metamath Proof Explorer

Theorem albiim

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

		Ref	Expression
	Assertion	albiim	$⊢ \forall x (φ \leftrightarrow ψ) \leftrightarrow (\forall x (φ \to ψ) \land \forall x (ψ \to φ))$

Step	Hyp	Ref	Expression
1		dfbi2	$⊢ (φ \leftrightarrow ψ) \leftrightarrow ((φ \to ψ) \land (ψ \to φ))$
2	1	albii	$⊢ \forall x (φ \leftrightarrow ψ) \leftrightarrow \forall x ((φ \to ψ) \land (ψ \to φ))$
3		19.26	$⊢ \forall x ((φ \to ψ) \land (ψ \to φ)) \leftrightarrow (\forall x (φ \to ψ) \land \forall x (ψ \to φ))$
4	2 3	bitri	$⊢ \forall x (φ \leftrightarrow ψ) \leftrightarrow (\forall x (φ \to ψ) \land \forall x (ψ \to φ))$