Theorem qseq1

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

	Ref	Expression
Assertion	qseq1	$$⊢A=B\to A/C=B/C$$

Proof

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