Theorem abssdv

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006) (Proof shortened by SN, 22-Dec-2024)

	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	ss2abdv	$$⊢\phi \to \left\{x|\psi \right\}\subseteq \left\{x|x\in A\right\}$$
3		abid1	$$⊢A=\left\{x|x\in A\right\}$$
4	2 3	sseqtrrdi	$$⊢\phi \to \left\{x|\psi \right\}\subseteq A$$