Theorem ax9vsep 4166

Description: Derive a weakened version of ax-9 1553, where x and y must be distinct, from Separation ax-sep 4161 and Extensionality ax-ext 2186. In intuitionistic logic a9evsep 4165 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

Step	Hyp	Ref	Expression
1		a9evsep 4165	. 2 |- E. x x = y
2		exalim 1524	. 2 |- ( E. x x = y -> -. A. x -. x = y )
3	1, 2	ax-mp 5	1 |- -. A. x -. x = y

Syntax hints:   -. wn 3   A.wal 1370   = wceq 1372   E.wex 1514

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 615  ax-in2 616  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556  ax-ext 2186  ax-sep 4161

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378

This theorem is referenced by: (None)