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Metamath Proof Explorer

Theorem dfint2

Description: Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998)

		Ref	Expression
	Assertion	dfint2	$⊢ ⋂ A = \{x \| \forall y \in A x \in y\}$

Step	Hyp	Ref	Expression
1		df-int	$⊢ ⋂ A = \{x \| \forall y (y \in A \to x \in y)\}$
2		df-ral	$⊢ \forall y \in A x \in y \leftrightarrow \forall y (y \in A \to x \in y)$
3	2	abbii	$⊢ \{x \| \forall y \in A x \in y\} = \{x \| \forall y (y \in A \to x \in y)\}$
4	1 3	eqtr4i	$⊢ ⋂ A = \{x \| \forall y \in A x \in y\}$