Theorem wl-clabtv 35776

Description: Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 35777. (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 360	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2	1	sbbidv 2077	. . 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 313	. 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 1537  [wsb 2062   ∈ wcel 2101  {cab 2710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2111  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725

This theorem is referenced by: (None)