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Metamath Proof Explorer

Theorem rabssdv

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015)

		Ref	Expression
	Hypothesis	rabssdv.1	$⊢ (φ \land x \in A \land ψ) \to x \in B$
	Assertion	rabssdv	$⊢ φ \to \{x \in A \| ψ\} \subseteq B$

Step	Hyp	Ref	Expression
1		rabssdv.1	$⊢ (φ \land x \in A \land ψ) \to x \in B$
2	1	3exp	$⊢ φ \to (x \in A \to (ψ \to x \in B))$
3	2	ralrimiv	$⊢ φ \to \forall x \in A (ψ \to x \in B)$
4		rabss	$⊢ \{x \in A \| ψ\} \subseteq B \leftrightarrow \forall x \in A (ψ \to x \in B)$
5	3 4	sylibr	$⊢ φ \to \{x \in A \| ψ\} \subseteq B$