Theorem qseq2

Description: Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995)

	Ref	Expression
Assertion	qseq2	$$⊢A=B\to C/A=C/B$$

Proof

Step	Hyp	Ref	Expression
1		eceq2	$$⊢A=B\to \left[x\right]A=\left[x\right]B$$
2	1	eqeq2d	$$⊢A=B\to \left(y=\left[x\right]A\leftrightarrow y=\left[x\right]B\right)$$
3	2	rexbidv	$$⊢A=B\to \left(\exists x\in Cy=\left[x\right]A\leftrightarrow \exists x\in Cy=\left[x\right]B\right)$$
4	3	abbidv	$$⊢A=B\to \left\{y|\exists x\in Cy=\left[x\right]A\right\}=\left\{y|\exists x\in Cy=\left[x\right]B\right\}$$
5		df-qs	$$⊢C/A=\left\{y|\exists x\in Cy=\left[x\right]A\right\}$$
6		df-qs	$$⊢C/B=\left\{y|\exists x\in Cy=\left[x\right]B\right\}$$
7	4 5 6	3eqtr4g	$$⊢A=B\to C/A=C/B$$