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Metamath Proof Explorer

Theorem rabbi2dva

Description: Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014)

		Ref	Expression
	Hypothesis	rabbi2dva.1	$⊢ (φ \land x \in A) \to (x \in B \leftrightarrow ψ)$
	Assertion	rabbi2dva	$⊢ φ \to A \cap B = \{x \in A \| ψ\}$

Step	Hyp	Ref	Expression
1		rabbi2dva.1	$⊢ (φ \land x \in A) \to (x \in B \leftrightarrow ψ)$
2		dfin5	$⊢ A \cap B = \{x \in A \| x \in B\}$
3	1	rabbidva	$⊢ φ \to \{x \in A \| x \in B\} = \{x \in A \| ψ\}$
4	2 3	eqtrid	$⊢ φ \to A \cap B = \{x \in A \| ψ\}$