Theorem cbvab

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Usage of the weaker cbvabw and cbvabv are preferred. (Contributed by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 16-Nov-2019) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvab.1	$$⊢Ⅎy\phi$$
2		cbvab.2	$$⊢Ⅎx\psi$$
3		cbvab.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1	sbco2	$$⊢\left[z/y\right]\left[y/x\right]\phi \leftrightarrow \left[z/x\right]\phi$$
5	2 3	sbie	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$
6	5	sbbii	$$⊢\left[z/y\right]\left[y/x\right]\phi \leftrightarrow \left[z/y\right]\psi$$
7	4 6	bitr3i	$$⊢\left[z/x\right]\phi \leftrightarrow \left[z/y\right]\psi$$
8		df-clab	$$⊢z\in \left\{x|\phi \right\}\leftrightarrow \left[z/x\right]\phi$$
9		df-clab	$$⊢z\in \left\{y|\psi \right\}\leftrightarrow \left[z/y\right]\psi$$
10	7 8 9	3bitr4i	$$⊢z\in \left\{x|\phi \right\}\leftrightarrow z\in \left\{y|\psi \right\}$$
11	10	eqriv	$$⊢\left\{x|\phi \right\}=\left\{y|\psi \right\}$$