ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbco3 Structured version   GIF version

Theorem sbco3 1845
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
Assertion
Ref Expression
sbco3 ([z / y][y / x]φ ↔ [z / x][x / y]φ)

Proof of Theorem sbco3
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 sbco3xzyz 1844 . . 3 ([w / y][y / x]φ ↔ [w / x][x / y]φ)
21sbbii 1645 . 2 ([z / w][w / y][y / x]φ ↔ [z / w][w / x][x / y]φ)
3 ax-17 1416 . . 3 ([y / x]φw[y / x]φ)
43sbco2h 1835 . 2 ([z / w][w / y][y / x]φ ↔ [z / y][y / x]φ)
5 ax-17 1416 . . 3 ([x / y]φw[x / y]φ)
65sbco2h 1835 . 2 ([z / w][w / x][x / y]φ ↔ [z / x][x / y]φ)
72, 4, 63bitr3i 199 1 ([z / y][y / x]φ ↔ [z / x][x / 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:  sbcom  1846
  Copyright terms: Public domain W3C validator