ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  clelsb1 GIF version

Theorem clelsb1 2298
Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2171). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
clelsb1 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1539 . . 3 𝑥 𝑤𝐴
21sbco2 1981 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑤]𝑤𝐴)
3 nfv 1539 . . . 4 𝑤 𝑥𝐴
4 eleq1 2256 . . . 4 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
53, 4sbie 1802 . . 3 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
65sbbii 1776 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑥]𝑥𝐴)
7 nfv 1539 . . 3 𝑤 𝑦𝐴
8 eleq1 2256 . . 3 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
97, 8sbie 1802 . 2 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
102, 6, 93bitr3i 210 1 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
Colors of variables: wff set class
Syntax hints:  wb 105  [wsb 1773  wcel 2164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2186  df-clel 2189
This theorem is referenced by:  hblem  2301  eqabdv  2322  nfraldya  2529  nfrexdya  2530  cbvreu  2724  sbcel1v  3048  rmo3  3077  setindel  4570  elirr  4573  en2lp  4586  zfregfr  4606  tfi  4614  bdcriota  15375
  Copyright terms: Public domain W3C validator