Theorem ss2rabdv

Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006)

	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	ralrimiva	$$⊢\phi \to \forall x\in A\left(\psi \to \chi \right)$$
3		ss2rab	$$⊢\left\{x\in A|\psi \right\}\subseteq \left\{x\in A|\chi \right\}\leftrightarrow \forall x\in A\left(\psi \to \chi \right)$$
4	2 3	sylibr	$$⊢\phi \to \left\{x\in A|\psi \right\}\subseteq \left\{x\in A|\chi \right\}$$