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

Theorem ax9vsep 3909
Description: Derive a weakened version of ax-9 1465, where  x and  y must be distinct, from Separation ax-sep 3904 and Extensionality ax-ext 2064. In intuitionistic logic a9evsep 3908 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax9vsep  |-  -.  A. x  -.  x  =  y
Distinct variable group:    x, y

Proof of Theorem ax9vsep
StepHypRef Expression
1 a9evsep 3908 . 2  |-  E. x  x  =  y
2 exalim 1432 . 2  |-  ( E. x  x  =  y  ->  -.  A. x  -.  x  =  y
)
31, 2ax-mp 7 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1283    = wceq 1285   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-ext 2064  ax-sep 3904
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator