Theorem ralbiOLD

Description: Obsolete version of ralbi as of 17-Jun-2023. (Contributed by NM, 6-Oct-2003) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfra1	$$⊢Ⅎx\forall x\in A\left(\phi \leftrightarrow \psi \right)$$
2		rspa	$$⊢\left(\forall x\in A\left(\phi \leftrightarrow \psi \right)\wedge x\in A\right)\to \left(\phi \leftrightarrow \psi \right)$$
3	1 2	ralbida	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\to \left(\forall x\in A\phi \leftrightarrow \forall x\in A\psi \right)$$