MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clelsb3f Structured version   Visualization version   GIF version

Theorem clelsb3f 2917
Description: Substitution applied to an atomic wff (class version of elsb3 2583). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
Hypothesis
Ref Expression
clelsb3f.1 𝑦𝐴
Assertion
Ref Expression
clelsb3f ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)

Proof of Theorem clelsb3f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 clelsb3f.1 . . . 4 𝑦𝐴
21nfcri 2907 . . 3 𝑦 𝑤𝐴
32sbco2 2562 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
4 nfv 1995 . . . 4 𝑤 𝑦𝐴
5 eleq1w 2833 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
64, 5sbie 2555 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
76sbbii 2056 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
8 nfv 1995 . . 3 𝑤 𝑥𝐴
9 eleq1w 2833 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
108, 9sbie 2555 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
113, 7, 103bitr3i 290 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 2049  wcel 2145  wnfc 2900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clel 2767  df-nfc 2902
This theorem is referenced by:  rmo3f  29674  suppss2f  29779  fmptdF  29796  disjdsct  29820  esumpfinvalf  30478
  Copyright terms: Public domain W3C validator