This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ineq1

Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993) (Proof shortened by SN, 20-Sep-2023)

		Ref	Expression
	Assertion	ineq1	$⊢ A = B \to A \cap C = B \cap C$

Step	Hyp	Ref	Expression
1		rabeq	$⊢ A = B \to \{x \in A \| x \in C\} = \{x \in B \| x \in C\}$
2		dfin5	$⊢ A \cap C = \{x \in A \| x \in C\}$
3		dfin5	$⊢ B \cap C = \{x \in B \| x \in C\}$
4	1 2 3	3eqtr4g	$⊢ A = B \to A \cap C = B \cap C$