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Metamath Proof Explorer

Theorem difeq1

Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997) (Proof shortened by Andrew Salmon, 26-Jun-2011)

		Ref	Expression
	Assertion	difeq1	$⊢ A = B \to A ∖ C = B ∖ C$

Step	Hyp	Ref	Expression
1		rabeq	$⊢ A = B \to \{x \in A \| \neg x \in C\} = \{x \in B \| \neg x \in C\}$
2		dfdif2	$⊢ A ∖ C = \{x \in A \| \neg x \in C\}$
3		dfdif2	$⊢ B ∖ C = \{x \in B \| \neg x \in C\}$
4	1 2 3	3eqtr4g	$⊢ A = B \to A ∖ C = B ∖ C$