Theorem ax9vsep 3880

Description: Derive a weakened version of ax-9 1424, where x and y must be distinct, from Separation ax-sep 3875 and Extensionality ax-ext 2022. In intuitionistic logic a9evsep 3879 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 3879	. 2 |- E. x x = y
2		exalim 1391	. 2 |- ( E. x x = y -> -. A. x -. x = y )
3	1, 2	ax-mp 7	1 |- -. A. x -. x = y

Syntax hints:   -. wn 3   A.wal 1241   = wceq 1243   E.wex 1381

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 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by: (None)