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

Theorem clelsb4 2140
Description: Substitution applied to an atomic wff (class version of elsb4 1850). (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 1418 . . 3 y A w
21sbco2 1836 . 2 ([x / y][y / w]A w ↔ [x / w]A w)
3 nfv 1418 . . . 4 w A y
4 eleq2 2098 . . . 4 (w = y → (A wA y))
53, 4sbie 1671 . . 3 ([y / w]A wA y)
65sbbii 1645 . 2 ([x / y][y / w]A w ↔ [x / y]A y)
7 nfv 1418 . . 3 w A x
8 eleq2 2098 . . 3 (w = x → (A wA x))
97, 8sbie 1671 . 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 1390  [wsb 1642
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033
This theorem is referenced by:  peano1  4260  peano2  4261
  Copyright terms: Public domain W3C validator