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Theorem equviniv 1880
Description: A specialized version of equvini 2195 with a distinct variable restriction. (Contributed by Wolf Lammen, 8-Sep-2018.)
Assertion
Ref Expression
equviniv  |-  ( x  =  y  ->  E. z
( x  =  z  /\  y  =  z ) )
Distinct variable groups:    x, z    y, z

Proof of Theorem equviniv
StepHypRef Expression
1 ax6evr 1867 . 2  |-  E. z 
y  =  z
2 equtr 1873 . . . 4  |-  ( x  =  y  ->  (
y  =  z  ->  x  =  z )
)
32ancrd 563 . . 3  |-  ( x  =  y  ->  (
y  =  z  -> 
( x  =  z  /\  y  =  z ) ) )
43eximdv 1772 . 2  |-  ( x  =  y  ->  ( E. z  y  =  z  ->  E. z ( x  =  z  /\  y  =  z ) ) )
51, 4mpi 20 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  y  =  z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   E.wex 1671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672
This theorem is referenced by:  equvin  1881  ax8  1910  ax9  1917  axc9lem1  2106  nfeqf  2152  bj-ssbequ2  31320
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