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Metamath Proof Explorer

Theorem ss2rabdv

Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006) Avoid axioms. (Revised by TM, 1-Feb-2026)

		Ref	Expression
	Hypothesis	ss2rabdv.1	$⊢ (φ \land x \in A) \to (ψ \to χ)$
	Assertion	ss2rabdv	$⊢ φ \to \{x \in A \| ψ\} \subseteq \{x \in A \| χ\}$

Proof

Step	Hyp	Ref	Expression
1		ss2rabdv.1	$⊢ (φ \land x \in A) \to (ψ \to χ)$
2	1	ralrimiva	$⊢ φ \to \forall x \in A (ψ \to χ)$
3	2	ss2rabd	$⊢ φ \to \{x \in A \| ψ\} \subseteq \{x \in A \| χ\}$