Theorem ralimdvv

Description: Deduction doubly quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-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	adantr	$$⊢\left(\phi \wedge \left(x\in A\wedge y\in B\right)\right)\to \left(\psi \to \chi \right)$$
3	2	ralimdvva	$$⊢\phi \to \left(\forall x\in A\forall y\in B\psi \to \forall x\in A\forall y\in B\chi \right)$$