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	⊢ Ⅎ 𝑥 𝜑
Assertion	wl-clabt	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑥 ∣ ( 𝜑 → 𝜓 ) } )

Proof

Step	Hyp	Ref	Expression
1		wl-clabt.nf	⊢ Ⅎ 𝑥 𝜑
2		biimt	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜑 → 𝜓 ) ) )
3	1 2	sbbid	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ) )
4		df-clab	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜓 } ↔ [ 𝑦 / 𝑥 ] 𝜓 )
5		df-clab	⊢ ( 𝑦 ∈ { 𝑥 ∣ ( 𝜑 → 𝜓 ) } ↔ [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) )
6	3 4 5	3bitr4g	⊢ ( 𝜑 → ( 𝑦 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝑦 ∈ { 𝑥 ∣ ( 𝜑 → 𝜓 ) } ) )
7	6	eqrdv	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑥 ∣ ( 𝜑 → 𝜓 ) } )