Theorem ralbida

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003) (Proof shortened by Wolf Lammen, 31-Oct-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ralbida.1	$$⊢Ⅎx\phi$$
2		ralbida.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	biimpd	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \to \chi \right)$$
4	1 3	ralimdaa	$$⊢\phi \to \left(\forall x\in A\psi \to \forall x\in A\chi \right)$$
5	2	biimprd	$$⊢\left(\phi \wedge x\in A\right)\to \left(\chi \to \psi \right)$$
6	1 5	ralimdaa	$$⊢\phi \to \left(\forall x\in A\chi \to \forall x\in A\psi \right)$$
7	4 6	impbid	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in A\chi \right)$$