Theorem ss2abdv

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011) (Revised by Steven Nguyen, 28-Jun-2024)

	Ref	Expression
Hypothesis	ss2abdv.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
Assertion	ss2abdv	$$⊢\phi \to \left\{x|\psi \right\}\subseteq \left\{x|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		ss2abdv.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2	1	sbimdv	$$⊢\phi \to \left(\left[y/x\right]\psi \to \left[y/x\right]\chi \right)$$
3		df-clab	$$⊢y\in \left\{x|\psi \right\}\leftrightarrow \left[y/x\right]\psi$$
4		df-clab	$$⊢y\in \left\{x|\chi \right\}\leftrightarrow \left[y/x\right]\chi$$
5	2 3 4	3imtr4g	$$⊢\phi \to \left(y\in \left\{x|\psi \right\}\to y\in \left\{x|\chi \right\}\right)$$
6	5	ssrdv	$$⊢\phi \to \left\{x|\psi \right\}\subseteq \left\{x|\chi \right\}$$