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	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \to \chi \right)$$
Assertion	ss2rabdv	$$⊢\phi \to \left\{x\in A|\psi \right\}\subseteq \left\{x\in A|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		ss2rabdv.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \to \chi \right)$$
2	1	imdistanda	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\to \left(x\in A\wedge \chi \right)\right)$$
3	2	ss2abdv	$$⊢\phi \to \left\{x|\left(x\in A\wedge \psi \right)\right\}\subseteq \left\{x|\left(x\in A\wedge \chi \right)\right\}$$
4		df-rab	$$⊢\left\{x\in A|\psi \right\}=\left\{x|\left(x\in A\wedge \psi \right)\right\}$$
5		df-rab	$$⊢\left\{x\in A|\chi \right\}=\left\{x|\left(x\in A\wedge \chi \right)\right\}$$
6	3 4 5	3sstr4g	$$⊢\phi \to \left\{x\in A|\psi \right\}\subseteq \left\{x\in A|\chi \right\}$$