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Metamath Proof Explorer

Theorem qseq1

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

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

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