Theorem csbconstg

Description: Substitution doesn't affect a constant B (in which x does not occur). csbconstgf with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012) Avoid ax-12 . (Revised by GG, 15-Oct-2024)

	Ref	Expression
Assertion	csbconstg	$$⊢A\in V\to ⦋A/x⦌B=B$$

Proof

Step	Hyp	Ref	Expression
1		csbeq1	$$⊢y=A\to ⦋y/x⦌B=⦋A/x⦌B$$
2	1	eqeq1d	$$⊢y=A\to \left(⦋y/x⦌B=B\leftrightarrow ⦋A/x⦌B=B\right)$$
3		df-csb	$$⊢⦋y/x⦌B=\left\{z|\underset{\dot{}}{[}y/x\underset{\dot{}}{]}z\in B\right\}$$
4		sbcg	$$⊢y\in \mathrm{V}\to \left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}z\in B\leftrightarrow z\in B\right)$$
5	4	elv	$$⊢\underset{\dot{}}{[}y/x\underset{\dot{}}{]}z\in B\leftrightarrow z\in B$$
6	5	abbii	$$⊢\left\{z|\underset{\dot{}}{[}y/x\underset{\dot{}}{]}z\in B\right\}=\left\{z|z\in B\right\}$$
7		abid2	$$⊢\left\{z|z\in B\right\}=B$$
8	3 6 7	3eqtri	$$⊢⦋y/x⦌B=B$$
9	2 8	vtoclg	$$⊢A\in V\to ⦋A/x⦌B=B$$