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	$$⊢x\in A\to \left(\phi \to \psi \right)$$
Assertion	ss2rabi	$$⊢\left\{x\in A|\phi \right\}\subseteq \left\{x\in A|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		ss2rabi.1	$$⊢x\in A\to \left(\phi \to \psi \right)$$
2	1	adantl	$$⊢\left(\top \wedge x\in A\right)\to \left(\phi \to \psi \right)$$
3	2	ss2rabdv	$$⊢\top \to \left\{x\in A|\phi \right\}\subseteq \left\{x\in A|\psi \right\}$$
4	3	mptru	$$⊢\left\{x\in A|\phi \right\}\subseteq \left\{x\in A|\psi \right\}$$