Theorem cbvraldva

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017) Avoid ax-9 , ax-ext . (Revised by Wolf Lammen, 9-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		cbvraldva.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	ancoms	$$⊢\left(x=y\wedge \phi \right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	pm5.74da	$$⊢x=y\to \left(\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)\right)$$
4	3	cbvralvw	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall y\in A\left(\phi \to \chi \right)$$
5		r19.21v	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\in A\psi \right)$$
6		r19.21v	$$⊢\forall y\in A\left(\phi \to \chi \right)\leftrightarrow \left(\phi \to \forall y\in A\chi \right)$$
7	4 5 6	3bitr3i	$$⊢\left(\phi \to \forall x\in A\psi \right)\leftrightarrow \left(\phi \to \forall y\in A\chi \right)$$
8	7	pm5.74ri	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall y\in A\chi \right)$$