ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  clelsb4 Structured version   GIF version

Theorem clelsb4 2125
Description: Substitution applied to an atomic wff (class version of elsb4 1835). (Contributed by Jim Kingdon, 22-Nov-2018.)
Assertion
Ref Expression
clelsb4 ([x / y]A yA x)
Distinct variable group:   y,A
Allowed substitution hint:   A(x)

Proof of Theorem clelsb4
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1402 . . 3 y A w
21sbco2 1821 . 2 ([x / y][y / w]A w ↔ [x / w]A w)
3 nfv 1402 . . . 4 w A y
4 eleq2 2083 . . . 4 (w = y → (A wA y))
53, 4sbie 1656 . . 3 ([y / w]A wA y)
65sbbii 1630 . 2 ([x / y][y / w]A w ↔ [x / y]A y)
7 nfv 1402 . . 3 w A x
8 eleq2 2083 . . 3 (w = x → (A wA x))
97, 8sbie 1656 . 2 ([x / w]A wA x)
102, 6, 93bitr3i 199 1 ([x / y]A yA x)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1374  [wsb 1627
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-cleq 2015  df-clel 2018
This theorem is referenced by:  peano1  4244  peano2  4245
  Copyright terms: Public domain W3C validator