Theorem wl-clabtv 34844

Description: Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 34845. (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 363	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2	1	sbbidv 2084	. . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥](𝜑 → 𝜓)))
3		df-clab 2800	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
4		df-clab 2800	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)} ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
5	2, 3, 4	3bitr4g 316	. 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)}))
6	5	eqrdv 2819	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})

Syntax hints:   → wi 4   = wceq 1537  [wsb 2069   ∈ wcel 2114  {cab 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814

This theorem is referenced by: (None)