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Theorem sb8ab 2175
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 1855 . . 3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3 df-clab 2043 . . 3 (𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
4 df-clab 2043 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
52, 3, 43bitr4ri 206 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑})
65eqriv 2053 1 {𝑥𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wnf 1365  wcel 1409  [wsb 1661  {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049
This theorem is referenced by: (None)
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