Theorem cbvabw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvab with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by Andrew Salmon, 11-Jul-2011) (Revised by Gino Giotto, 23-May-2024)

	Ref	Expression
Hypotheses	cbvabw.1	$$⊢Ⅎy\phi$$
	cbvabw.2	$$⊢Ⅎx\psi$$
	cbvabw.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvabw	$$⊢\left\{x|\phi \right\}=\left\{y|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		cbvabw.1	$$⊢Ⅎy\phi$$
2		cbvabw.2	$$⊢Ⅎx\psi$$
3		cbvabw.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4		nfv	$$⊢Ⅎyx=w$$
5	4 1	nfim	$$⊢Ⅎy\left(x=w\to \phi \right)$$
6		nfv	$$⊢Ⅎxy=w$$
7	6 2	nfim	$$⊢Ⅎx\left(y=w\to \psi \right)$$
8		equequ1	$$⊢x=y\to \left(x=w\leftrightarrow y=w\right)$$
9	8 3	imbi12d	$$⊢x=y\to \left(\left(x=w\to \phi \right)\leftrightarrow \left(y=w\to \psi \right)\right)$$
10	5 7 9	cbvalv1	$$⊢\forall x\left(x=w\to \phi \right)\leftrightarrow \forall y\left(y=w\to \psi \right)$$
11	10	imbi2i	$$⊢\left(w=z\to \forall x\left(x=w\to \phi \right)\right)\leftrightarrow \left(w=z\to \forall y\left(y=w\to \psi \right)\right)$$
12	11	albii	$$⊢\forall w\left(w=z\to \forall x\left(x=w\to \phi \right)\right)\leftrightarrow \forall w\left(w=z\to \forall y\left(y=w\to \psi \right)\right)$$
13		df-sb	$$⊢\left[z/x\right]\phi \leftrightarrow \forall w\left(w=z\to \forall x\left(x=w\to \phi \right)\right)$$
14		df-sb	$$⊢\left[z/y\right]\psi \leftrightarrow \forall w\left(w=z\to \forall y\left(y=w\to \psi \right)\right)$$
15	12 13 14	3bitr4i	$$⊢\left[z/x\right]\phi \leftrightarrow \left[z/y\right]\psi$$
16		df-clab	$$⊢z\in \left\{x|\phi \right\}\leftrightarrow \left[z/x\right]\phi$$
17		df-clab	$$⊢z\in \left\{y|\psi \right\}\leftrightarrow \left[z/y\right]\psi$$
18	15 16 17	3bitr4i	$$⊢z\in \left\{x|\phi \right\}\leftrightarrow z\in \left\{y|\psi \right\}$$
19	18	eqriv	$$⊢\left\{x|\phi \right\}=\left\{y|\psi \right\}$$