Theorem ralbid

Description: Formula-building rule for restricted universal quantifier (deduction form). For a version based on fewer axioms see ralbidv . (Contributed by NM, 27-Jun-1998)

	Ref	Expression
Hypotheses	ralbid.1	$$⊢Ⅎx\phi$$
	ralbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	ralbid	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralbid.1	$$⊢Ⅎx\phi$$
2		ralbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	2	adantr	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
4	1 3	ralbida	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in A\chi \right)$$