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

Step	Hyp	Ref	Expression
1		a9evsep 3908	. 2 |- E. x x = y
2		exalim 1432	. 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 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)