Theorem ralimdvv

Description: Deduction doubly quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-2025) Shorten and reduce DV conditions. (Revised by Eric Schmidt, 18-Nov-2025)

	Ref	Expression
Hypothesis	ralimdvv.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
Assertion	ralimdvv	$$⊢\phi \to \left(\forall x\in A\forall y\in B\psi \to \forall x\in A\forall y\in B\chi \right)$$

Proof

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