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Metamath Proof Explorer

Theorem ss2rabi

Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999) Avoid axioms. (Revised by SN, 4-Feb-2025)

		Ref	Expression
	Hypothesis	ss2rabi.1	$⊢ x \in A \to (φ \to ψ)$
	Assertion	ss2rabi	$⊢ \{x \in A \| φ\} \subseteq \{x \in A \| ψ\}$

Proof

Step	Hyp	Ref	Expression
1		ss2rabi.1	$⊢ x \in A \to (φ \to ψ)$
2	1	adantl	$⊢ (⊤ \land x \in A) \to (φ \to ψ)$
3	2	ss2rabdv	$⊢ ⊤ \to \{x \in A \| φ\} \subseteq \{x \in A \| ψ\}$
4	3	mptru	$⊢ \{x \in A \| φ\} \subseteq \{x \in A \| ψ\}$