Theorem cbvaldvaw

Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017) (Revised by Gino Giotto, 10-Jan-2024) Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf Lammen, 10-Feb-2024)

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

Proof

Step	Hyp	Ref	Expression
1		cbvaldvaw.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	cbvalvw	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \forall y\left(\phi \to \chi \right)$$
5		19.21v	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$
6		19.21v	$$⊢\forall y\left(\phi \to \chi \right)\leftrightarrow \left(\phi \to \forall y\chi \right)$$
7	4 5 6	3bitr3i	$$⊢\left(\phi \to \forall x\psi \right)\leftrightarrow \left(\phi \to \forall y\chi \right)$$
8	7	pm5.74ri	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$