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Theorem ax9vsep 4021
Description: Derive a weakened version of ax-9 1496, where  x and  y must be distinct, from Separation ax-sep 4016 and Extensionality ax-ext 2099. In intuitionistic logic a9evsep 4020 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 4020 . 2  |-  E. x  x  =  y
2 exalim 1463 . 2  |-  ( E. x  x  =  y  ->  -.  A. x  -.  x  =  y
)
31, 2ax-mp 5 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1314    = wceq 1316   E.wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322
This theorem is referenced by: (None)
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