Theorem ralbii2

Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005)

	Ref	Expression
Hypothesis	ralbii2.1	$$⊢\left(x\in A\to \phi \right)\leftrightarrow \left(x\in B\to \psi \right)$$
Assertion	ralbii2	$$⊢\forall x\in A\phi \leftrightarrow \forall x\in B\psi$$

Proof

Step	Hyp	Ref	Expression
1		ralbii2.1	$$⊢\left(x\in A\to \phi \right)\leftrightarrow \left(x\in B\to \psi \right)$$
2	1	albii	$$⊢\forall x\left(x\in A\to \phi \right)\leftrightarrow \forall x\left(x\in B\to \psi \right)$$
3		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
4		df-ral	$$⊢\forall x\in B\psi \leftrightarrow \forall x\left(x\in B\to \psi \right)$$
5	2 3 4	3bitr4i	$$⊢\forall x\in A\phi \leftrightarrow \forall x\in B\psi$$