Theorem 3ralbidv

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

	Ref	Expression
Hypothesis	3ralbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	3ralbidv	$$⊢\phi \to \left(\forall x\in A\forall y\in B\forall z\in C\psi \leftrightarrow \forall x\in A\forall y\in B\forall z\in C\chi \right)$$

Proof

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