ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb8ab GIF version

Theorem sb8ab 2261
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 1938 . . 3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3 df-clab 2126 . . 3 (𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
4 df-clab 2126 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
52, 3, 43bitr4ri 212 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑})
65eqriv 2136 1 {𝑥𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wnf 1436  wcel 1480  [wsb 1735  {cab 2125
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator