ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbco4lem Structured version   GIF version

Theorem sbco4lem 1879
Description: Lemma for sbco4 1880. It replaces the temporary variable v with another temporary variable w. (Contributed by Jim Kingdon, 26-Sep-2018.)
Assertion
Ref Expression
sbco4lem ([x / v][y / x][v / y]φ ↔ [x / w][y / x][w / y]φ)
Distinct variable groups:   w,v,φ   x,v,w   y,v,w
Allowed substitution hints:   φ(x,y)

Proof of Theorem sbco4lem
StepHypRef Expression
1 sbcom2 1860 . . 3 ([w / v][y / x][v / w][w / y]φ ↔ [y / x][w / v][v / w][w / y]φ)
21sbbii 1645 . 2 ([x / w][w / v][y / x][v / w][w / y]φ ↔ [x / w][y / x][w / v][v / w][w / y]φ)
3 nfv 1418 . . . . . . 7 wφ
43sbco2 1836 . . . . . 6 ([v / w][w / y]φ ↔ [v / y]φ)
54sbbii 1645 . . . . 5 ([y / x][v / w][w / y]φ ↔ [y / x][v / y]φ)
65sbbii 1645 . . . 4 ([w / v][y / x][v / w][w / y]φ ↔ [w / v][y / x][v / y]φ)
76sbbii 1645 . . 3 ([x / w][w / v][y / x][v / w][w / y]φ ↔ [x / w][w / v][y / x][v / y]φ)
8 nfv 1418 . . . 4 w[y / x][v / y]φ
98sbco2 1836 . . 3 ([x / w][w / v][y / x][v / y]φ ↔ [x / v][y / x][v / y]φ)
107, 9bitri 173 . 2 ([x / w][w / v][y / x][v / w][w / y]φ ↔ [x / v][y / x][v / y]φ)
11 nfv 1418 . . . . 5 v[w / y]φ
1211sbid2 1727 . . . 4 ([w / v][v / w][w / y]φ ↔ [w / y]φ)
1312sbbii 1645 . . 3 ([y / x][w / v][v / w][w / y]φ ↔ [y / x][w / y]φ)
1413sbbii 1645 . 2 ([x / w][y / x][w / v][v / w][w / y]φ ↔ [x / w][y / x][w / y]φ)
152, 10, 143bitr3i 199 1 ([x / v][y / x][v / y]φ ↔ [x / w][y / x][w / y]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  [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
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  sbco4  1880
  Copyright terms: Public domain W3C validator