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

Theorem sb8ab 2141
 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 1821 . . 3 ([z / y][y / x]φ ↔ [z / x]φ)
3 df-clab 2009 . . 3 (z {y ∣ [y / x]φ} ↔ [z / y][y / x]φ)
4 df-clab 2009 . . 3 (z {xφ} ↔ [z / x]φ)
52, 3, 43bitr4ri 202 . 2 (z {xφ} ↔ z {y ∣ [y / x]φ})
65eqriv 2019 1 {xφ} = {y ∣ [y / x]φ}
 Colors of variables: wff set class Syntax hints:   = wceq 1228  Ⅎwnf 1329   ∈ wcel 1374  [wsb 1627  {cab 2008 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator