Theorem albi

Description: Theorem 19.15 of Margaris p. 90. (Contributed by NM, 24-Jan-1993)

	Ref	Expression
Assertion	albi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\forall x\phi \leftrightarrow \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		biimp	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\phi \to \psi \right)$$
2	1	al2imi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\forall x\phi \to \forall x\psi \right)$$
3		biimpr	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \phi \right)$$
4	3	al2imi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\forall x\psi \to \forall x\phi \right)$$
5	2 4	impbid	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\forall x\phi \leftrightarrow \forall x\psi \right)$$