Theorem wl-clabt

Description: Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv . (Contributed by Wolf Lammen, 29-May-2023)

	Ref	Expression
Hypothesis	wl-clabt.nf	$$⊢Ⅎx\phi$$
Assertion	wl-clabt	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\left(\phi \to \psi \right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		wl-clabt.nf	$$⊢Ⅎx\phi$$
2		biimt	$$⊢\phi \to \left(\psi \leftrightarrow \left(\phi \to \psi \right)\right)$$
3	1 2	sbbid	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \left[y/x\right]\left(\phi \to \psi \right)\right)$$
4		df-clab	$$⊢y\in \left\{x|\psi \right\}\leftrightarrow \left[y/x\right]\psi$$
5		df-clab	$$⊢y\in \left\{x|\left(\phi \to \psi \right)\right\}\leftrightarrow \left[y/x\right]\left(\phi \to \psi \right)$$
6	3 4 5	3bitr4g	$$⊢\phi \to \left(y\in \left\{x|\psi \right\}\leftrightarrow y\in \left\{x|\left(\phi \to \psi \right)\right\}\right)$$
7	6	eqrdv	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\left(\phi \to \psi \right)\right\}$$