Theorem ssabdv

Description: Deduction of abstraction subclass from implication. (Contributed by SN, 22-Dec-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ssabdv.1	$$⊢\phi \to \left(x\in A\to \psi \right)$$
2		abid1	$$⊢A=\left\{x|x\in A\right\}$$
3	1	ss2abdv	$$⊢\phi \to \left\{x|x\in A\right\}\subseteq \left\{x|\psi \right\}$$
4	2 3	eqsstrid	$$⊢\phi \to A\subseteq \left\{x|\psi \right\}$$