Theorem ralimdvva

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90 ( alim ). (Contributed by AV, 27-Nov-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralimdvva.1	$$⊢\left(\phi \wedge \left(x\in A\wedge y\in B\right)\right)\to \left(\psi \to \chi \right)$$
2	1	anassrs	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge y\in B\right)\to \left(\psi \to \chi \right)$$
3	2	ralimdva	$$⊢\left(\phi \wedge x\in A\right)\to \left(\forall y\in B\psi \to \forall y\in B\chi \right)$$
4	3	ralimdva	$$⊢\phi \to \left(\forall x\in A\forall y\in B\psi \to \forall x\in A\forall y\in B\chi \right)$$