Theorem abssdv

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006)

	Ref	Expression
Hypothesis	abssdv.1	$$⊢\phi \to \left(\psi \to x\in A\right)$$
Assertion	abssdv	$$⊢\phi \to \left\{x|\psi \right\}\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		abssdv.1	$$⊢\phi \to \left(\psi \to x\in A\right)$$
2	1	alrimiv	$$⊢\phi \to \forall x\left(\psi \to x\in A\right)$$
3		abss	$$⊢\left\{x|\psi \right\}\subseteq A\leftrightarrow \forall x\left(\psi \to x\in A\right)$$
4	2 3	sylibr	$$⊢\phi \to \left\{x|\psi \right\}\subseteq A$$