Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax9vsep Structured version   GIF version

Theorem ax9vsep 3854
 Description: Derive a weakened version of ax-9 1405, where x and y must be distinct, from Separation ax-sep 3849 and Extensionality ax-ext 2004. In intuitionistic logic a9evsep 3853 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax9vsep ¬ x ¬ x = y
Distinct variable group:   x,y

Proof of Theorem ax9vsep
StepHypRef Expression
1 a9evsep 3853 . 2 x x = y
2 exalim 1372 . 2 (x x = y → ¬ x ¬ x = y)
31, 2ax-mp 7 1 ¬ x ¬ x = y
 Colors of variables: wff set class Syntax hints:  ¬ wn 3  ∀wal 1226   = wceq 1228  ∃wex 1362 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 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-ial 1409  ax-ext 2004  ax-sep 3849 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator