Theorem ss2ab1

Description: Class abstractions in a subclass relationship, closed form. One direction of ss2ab using fewer axioms. (Contributed by SN, 22-Dec-2024)

	Ref	Expression
Assertion	ss2ab1	$$⊢\forall x\left(\phi \to \psi \right)\to \left\{x|\phi \right\}\subseteq \left\{x|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		spsbim	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\left[t/x\right]\phi \to \left[t/x\right]\psi \right)$$
2		df-clab	$$⊢t\in \left\{x|\phi \right\}\leftrightarrow \left[t/x\right]\phi$$
3		df-clab	$$⊢t\in \left\{x|\psi \right\}\leftrightarrow \left[t/x\right]\psi$$
4	1 2 3	3imtr4g	$$⊢\forall x\left(\phi \to \psi \right)\to \left(t\in \left\{x|\phi \right\}\to t\in \left\{x|\psi \right\}\right)$$
5	4	ssrdv	$$⊢\forall x\left(\phi \to \psi \right)\to \left\{x|\phi \right\}\subseteq \left\{x|\psi \right\}$$