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Theorem wl-clabtv 36454
Description: Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 36455. (Contributed by Wolf Lammen, 29-May-2023.)
Assertion
Ref Expression
wl-clabtv (𝜑 → {𝑥𝜓} = {𝑥 ∣ (𝜑𝜓)})
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem wl-clabtv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 biimt 360 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
21sbbidv 2082 . . 3 (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥](𝜑𝜓)))
3 df-clab 2710 . . 3 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
4 df-clab 2710 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑𝜓)} ↔ [𝑦 / 𝑥](𝜑𝜓))
52, 3, 43bitr4g 313 . 2 (𝜑 → (𝑦 ∈ {𝑥𝜓} ↔ 𝑦 ∈ {𝑥 ∣ (𝜑𝜓)}))
65eqrdv 2730 1 (𝜑 → {𝑥𝜓} = {𝑥 ∣ (𝜑𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  [wsb 2067  wcel 2106  {cab 2709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724
This theorem is referenced by: (None)
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