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Metamath Proof Explorer

Theorem ssrdv

Description: Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995)

		Ref	Expression
	Hypothesis	ssrdv.1	$⊢ φ \to (x \in A \to x \in B)$
	Assertion	ssrdv	$⊢ φ \to A \subseteq B$

Step	Hyp	Ref	Expression
1		ssrdv.1	$⊢ φ \to (x \in A \to x \in B)$
2	1	alrimiv	$⊢ φ \to \forall x (x \in A \to x \in B)$
3		df-ss	$⊢ A \subseteq B \leftrightarrow \forall x (x \in A \to x \in B)$
4	2 3	sylibr	$⊢ φ \to A \subseteq B$