Theorem wl-clabtv 36396

Description: Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 36397. (Contributed by Wolf Lammen, 29-May-2023.)

Assertion

Ref	Expression
wl-clabtv	⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem wl-clabtv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biimt 361	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2	1	sbbidv 2083	. . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥](𝜑 → 𝜓)))
3		df-clab 2711	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
4		df-clab 2711	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)} ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
5	2, 3, 4	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)}))
6	5	eqrdv 2731	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})

Syntax hints:   → wi 4   = wceq 1542  [wsb 2068   ∈ wcel 2107  {cab 2710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725

This theorem is referenced by: (None)