Theorem cbvraldva2

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	cbvraldva2.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
	cbvraldva2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
Assertion	cbvraldva2	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall y\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvraldva2.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		cbvraldva2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
3		simpr	$$⊢\left(\phi \wedge x=y\right)\to x=y$$
4	3 2	eleq12d	$$⊢\left(\phi \wedge x=y\right)\to \left(x\in A\leftrightarrow y\in B\right)$$
5	4 1	imbi12d	$$⊢\left(\phi \wedge x=y\right)\to \left(\left(x\in A\to \psi \right)\leftrightarrow \left(y\in B\to \chi \right)\right)$$
6	5	cbvaldvaw	$$⊢\phi \to \left(\forall x\left(x\in A\to \psi \right)\leftrightarrow \forall y\left(y\in B\to \chi \right)\right)$$
7		df-ral	$$⊢\forall x\in A\psi \leftrightarrow \forall x\left(x\in A\to \psi \right)$$
8		df-ral	$$⊢\forall y\in B\chi \leftrightarrow \forall y\left(y\in B\to \chi \right)$$
9	6 7 8	3bitr4g	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall y\in B\chi \right)$$