Theorem 4ralbidv

Description: Formula-building rule for restricted universal quantifiers (deduction form.) (Contributed by Scott Fenton, 20-Feb-2025)

	Ref	Expression
Hypothesis	4ralbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	4ralbidv	$$⊢\phi \to \left(\forall x\in A\forall y\in B\forall z\in C\forall w\in D\psi \leftrightarrow \forall x\in A\forall y\in B\forall z\in C\forall w\in D\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		4ralbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	ralbidv	$$⊢\phi \to \left(\forall w\in D\psi \leftrightarrow \forall w\in D\chi \right)$$
3	2	3ralbidv	$$⊢\phi \to \left(\forall x\in A\forall y\in B\forall z\in C\forall w\in D\psi \leftrightarrow \forall x\in A\forall y\in B\forall z\in C\forall w\in D\chi \right)$$