ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elsb4 Structured version   GIF version

Theorem elsb4 1850
Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb4 ([x / y]z yz x)
Distinct variable group:   y,z

Proof of Theorem elsb4
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 ax-17 1416 . . . . 5 (z yw z y)
2 elequ2 1598 . . . . 5 (w = y → (z wz y))
31, 2sbieh 1670 . . . 4 ([y / w]z wz y)
43sbbii 1645 . . 3 ([x / y][y / w]z w ↔ [x / y]z y)
5 ax-17 1416 . . . 4 (z wy z w)
65sbco2h 1835 . . 3 ([x / y][y / w]z w ↔ [x / w]z w)
74, 6bitr3i 175 . 2 ([x / y]z y ↔ [x / w]z w)
8 equsb1 1665 . . . 4 [x / w]w = x
9 elequ2 1598 . . . . 5 (w = x → (z wz x))
109sbimi 1644 . . . 4 ([x / w]w = x → [x / w](z wz x))
118, 10ax-mp 7 . . 3 [x / w](z wz x)
12 sbbi 1830 . . 3 ([x / w](z wz x) ↔ ([x / w]z w ↔ [x / w]z x))
1311, 12mpbi 133 . 2 ([x / w]z w ↔ [x / w]z x)
14 ax-17 1416 . . 3 (z xw z x)
1514sbh 1656 . 2 ([x / w]z xz x)
167, 13, 153bitri 195 1 ([x / y]z yz x)
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-14 1402  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:  peano2  4261
  Copyright terms: Public domain W3C validator