Theorem wl-clabtv

Description: Using class abstraction in a context, requiring x and ph disjoint, but based on fewer axioms than wl-clabt . (Contributed by Wolf Lammen, 29-May-2023)

	Ref	Expression
Assertion	wl-clabtv	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\left(\phi \to \psi \right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		biimt	$$⊢\phi \to \left(\psi \leftrightarrow \left(\phi \to \psi \right)\right)$$
2	1	sbbidv	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \left[y/x\right]\left(\phi \to \psi \right)\right)$$
3		df-clab	$$⊢y\in \left\{x|\psi \right\}\leftrightarrow \left[y/x\right]\psi$$
4		df-clab	$$⊢y\in \left\{x|\left(\phi \to \psi \right)\right\}\leftrightarrow \left[y/x\right]\left(\phi \to \psi \right)$$
5	2 3 4	3bitr4g	$$⊢\phi \to \left(y\in \left\{x|\psi \right\}\leftrightarrow y\in \left\{x|\left(\phi \to \psi \right)\right\}\right)$$
6	5	eqrdv	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\left(\phi \to \psi \right)\right\}$$