Theorem ralbidva

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 4-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 29-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralbidva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	pm5.74da	$$⊢\phi \to \left(\left(x\in A\to \psi \right)\leftrightarrow \left(x\in A\to \chi \right)\right)$$
3	2	ralbidv2	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in A\chi \right)$$