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Theorem sb8ab 2239
Description: Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
Hypothesis
Ref Expression
sb8ab.1 𝑦𝜑
Assertion
Ref Expression
sb8ab {𝑥𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}

Proof of Theorem sb8ab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sb8ab.1 . . . 4 𝑦𝜑
21sbco2 1916 . . 3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3 df-clab 2104 . . 3 (𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
4 df-clab 2104 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
52, 3, 43bitr4ri 212 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑})
65eqriv 2114 1 {𝑥𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wnf 1421  wcel 1465  [wsb 1720  {cab 2103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110
This theorem is referenced by: (None)
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