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	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑥 ∣ ( 𝜑 → 𝜓 ) } )

Proof

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