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Theorem a9evsep 3907
Description: Derive a weakened version of ax-i9 1439, where  x and  y must be distinct, from Separation ax-sep 3903 and Extensionality ax-ext 2038. The theorem  -.  A. x -.  x  =  y also holds (ax9vsep 3908), but in intuitionistic logic  E. x x  =  y is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a9evsep  |-  E. x  x  =  y
Distinct variable group:    x, y

Proof of Theorem a9evsep
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax-sep 3903 . 2  |-  E. x A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )
2 id 19 . . . . . . . 8  |-  ( z  =  z  ->  z  =  z )
32biantru 290 . . . . . . 7  |-  ( z  e.  y  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )
43bibi2i 220 . . . . . 6  |-  ( ( z  e.  x  <->  z  e.  y )  <->  ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) ) )
54biimpri 128 . . . . 5  |-  ( ( z  e.  x  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )  ->  (
z  e.  x  <->  z  e.  y ) )
65alimi 1360 . . . 4  |-  ( A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )  ->  A. z
( z  e.  x  <->  z  e.  y ) )
7 ax-ext 2038 . . . 4  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
86, 7syl 14 . . 3  |-  ( A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )  ->  x  =  y )
98eximi 1507 . 2  |-  ( E. x A. z ( z  e.  x  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )  ->  E. x  x  =  y )
101, 9ax-mp 7 1  |-  E. x  x  =  y
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  ax9vsep  3908
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