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Theorem ax9vsep 4145
Description: Derive a weakened version of ax9 1889 ( i.e. ax9v 1636), where  x and  y must be distinct, from Separation ax-sep 4141 and Extensionality ax-ext 2264. See ax9 1889 for the derivation of ax9 1889 from ax9v 1636. (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
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax-sep 4141 . . 3  |-  E. x A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )
2 id 19 . . . . . . . . 9  |-  ( z  =  z  ->  z  =  z )
32biantru 491 . . . . . . . 8  |-  ( z  e.  y  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )
43bibi2i 304 . . . . . . 7  |-  ( ( z  e.  x  <->  z  e.  y )  <->  ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) ) )
54biimpri 197 . . . . . 6  |-  ( ( z  e.  x  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )  ->  (
z  e.  x  <->  z  e.  y ) )
65alimi 1546 . . . . 5  |-  ( A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )  ->  A. z
( z  e.  x  <->  z  e.  y ) )
7 ax-ext 2264 . . . . 5  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
86, 7syl 15 . . . 4  |-  ( A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )  ->  x  =  y )
98eximi 1563 . . 3  |-  ( E. x A. z ( z  e.  x  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )  ->  E. x  x  =  y )
101, 9ax-mp 8 . 2  |-  E. x  x  =  y
11 df-ex 1529 . 2  |-  ( E. x  x  =  y  <->  -.  A. x  -.  x  =  y )
1210, 11mpbi 199 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684
This theorem is referenced by:  ax9sep  29160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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