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Theorem ax9vsep 4126
Description: Derive a weakened version of ax-9 1531, where  x and  y must be distinct, from Separation ax-sep 4121 and Extensionality ax-ext 2159. In intuitionistic logic a9evsep 4125 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 4125 . 2  |-  E. x  x  =  y
2 exalim 1502 . 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 1351    = wceq 1353   E.wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534  ax-ext 2159  ax-sep 4121
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359
This theorem is referenced by: (None)
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