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Theorem ax6vsep 4564
Description: Derive a weakened version of ax-6 1752 ( i.e. ax6v 1753), where  x and  y must be distinct, from Separation ax-sep 4560 and Extensionality ax-ext 2432. See ax6 2008 for the derivation of ax-6 1752 from ax6v 1753. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6vsep  |-  -.  A. x  -.  x  =  y
Distinct variable group:    x, y

Proof of Theorem ax6vsep
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax-sep 4560 . . 3  |-  E. x A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )
2 id 22 . . . . . . . . 9  |-  ( z  =  z  ->  z  =  z )
32biantru 503 . . . . . . . 8  |-  ( z  e.  y  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )
43bibi2i 311 . . . . . . 7  |-  ( ( z  e.  x  <->  z  e.  y )  <->  ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) ) )
54biimpri 206 . . . . . 6  |-  ( ( z  e.  x  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )  ->  (
z  e.  x  <->  z  e.  y ) )
65alimi 1638 . . . . 5  |-  ( A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )  ->  A. z
( z  e.  x  <->  z  e.  y ) )
7 ax-ext 2432 . . . . 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 1661 . . 3  |-  ( E. x A. z ( z  e.  x  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )  ->  E. x  x  =  y )
101, 9ax-mp 5 . 2  |-  E. x  x  =  y
11 df-ex 1618 . 2  |-  ( E. x  x  =  y  <->  -.  A. x  -.  x  =  y )
1210, 11mpbi 208 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618
This theorem is referenced by: (None)
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