Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb4 Structured version   GIF version

Theorem sb4 1710
 Description: One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sb4 x x = y → ([y / x]φx(x = yφ)))

Proof of Theorem sb4
StepHypRef Expression
1 sb1 1646 . 2 ([y / x]φx(x = y φ))
2 equs5 1707 . 2 x x = y → (x(x = y φ) → x(x = yφ)))
31, 2syl5 28 1 x x = y → ([y / x]φx(x = yφ)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1240  ∃wex 1378  [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-in1 544  ax-in2 545  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-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643 This theorem is referenced by:  sb4b  1712  hbsb2  1714
 Copyright terms: Public domain W3C validator