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Theorem a9evsep 4018
Description: Derive a weakened version of ax-i9 1493, where  x and  y must be distinct, from Separation ax-sep 4014 and Extensionality ax-ext 2097. The theorem  -.  A. x -.  x  =  y also holds (ax9vsep 4019), 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 4014 . 2  |-  E. x A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )
2 id 19 . . . . . . . 8  |-  ( z  =  z  ->  z  =  z )
32biantru 298 . . . . . . 7  |-  ( z  e.  y  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )
43bibi2i 226 . . . . . 6  |-  ( ( z  e.  x  <->  z  e.  y )  <->  ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) ) )
54biimpri 132 . . . . 5  |-  ( ( z  e.  x  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )  ->  (
z  e.  x  <->  z  e.  y ) )
65alimi 1414 . . . 4  |-  ( A. z ( z  e.  x  <->  ( z  e.  y  /\  ( z  =  z  ->  z  =  z ) ) )  ->  A. z
( z  e.  x  <->  z  e.  y ) )
7 ax-ext 2097 . . . 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 1562 . 2  |-  ( E. x A. z ( z  e.  x  <->  ( z  e.  y  /\  (
z  =  z  -> 
z  =  z ) ) )  ->  E. x  x  =  y )
101, 9ax-mp 5 1  |-  E. x  x  =  y
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497  ax-ext 2097  ax-sep 4014
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  ax9vsep  4019
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