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Theorem ax9vsep 4336
Description: Derive a weakened version of ax-9 1667 ( i.e. ax9v 1668), where  x and  y must be distinct, from Separation ax-sep 4332 and Extensionality ax-ext 2419. See ax9 1954 for the derivation of ax-9 1667 from ax9v 1668. (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 4332 . . 3  |-  E. x A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )
2 id 21 . . . . . . . . 9  |-  ( z  =  z  ->  z  =  z )
32biantru 493 . . . . . . . 8  |-  ( z  e.  y  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )
43bibi2i 306 . . . . . . 7  |-  ( ( z  e.  x  <->  z  e.  y )  <->  ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) ) )
54biimpri 199 . . . . . 6  |-  ( ( z  e.  x  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )  ->  (
z  e.  x  <->  z  e.  y ) )
65alimi 1569 . . . . 5  |-  ( A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )  ->  A. z
( z  e.  x  <->  z  e.  y ) )
7 ax-ext 2419 . . . . 5  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
86, 7syl 16 . . . 4  |-  ( A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )  ->  x  =  y )
98eximi 1586 . . 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 1552 . 2  |-  ( E. x  x  =  y  <->  -.  A. x  -.  x  =  y )
1210, 11mpbi 201 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-ext 2419  ax-sep 4332
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
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