Theorem ralbiim

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

	Ref	Expression
Assertion	ralbiim	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\forall x\in A\left(\phi \to \psi \right)\wedge \forall x\in A\left(\psi \to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfbi2	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left(\phi \to \psi \right)\wedge \left(\psi \to \phi \right)\right)$$
2	1	ralbii	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\leftrightarrow \forall x\in A\left(\left(\phi \to \psi \right)\wedge \left(\psi \to \phi \right)\right)$$
3		r19.26	$$⊢\forall x\in A\left(\left(\phi \to \psi \right)\wedge \left(\psi \to \phi \right)\right)\leftrightarrow \left(\forall x\in A\left(\phi \to \psi \right)\wedge \forall x\in A\left(\psi \to \phi \right)\right)$$
4	2 3	bitri	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\forall x\in A\left(\phi \to \psi \right)\wedge \forall x\in A\left(\psi \to \phi \right)\right)$$