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Theorem wl-clabt 34877
Description: Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 34876. (Contributed by Wolf Lammen, 29-May-2023.)
Hypothesis
Ref Expression
wl-clabt.nf 𝑥𝜑
Assertion
Ref Expression
wl-clabt (𝜑 → {𝑥𝜓} = {𝑥 ∣ (𝜑𝜓)})

Proof of Theorem wl-clabt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 wl-clabt.nf . . . 4 𝑥𝜑
2 biimt 364 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
31, 2sbbid 2247 . . 3 (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥](𝜑𝜓)))
4 df-clab 2800 . . 3 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
5 df-clab 2800 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑𝜓)} ↔ [𝑦 / 𝑥](𝜑𝜓))
63, 4, 53bitr4g 317 . 2 (𝜑 → (𝑦 ∈ {𝑥𝜓} ↔ 𝑦 ∈ {𝑥 ∣ (𝜑𝜓)}))
76eqrdv 2819 1 (𝜑 → {𝑥𝜓} = {𝑥 ∣ (𝜑𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wnf 1785  [wsb 2070  wcel 2115  {cab 2799
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 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814
This theorem is referenced by: (None)
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