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Theorem equviniv 1857
Description: A specialized version of equvini 2146 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 ax6ev 1800 . 2  |-  E. z 
z  =  y
2 equcomi 1847 . . . 4  |-  ( z  =  y  ->  y  =  z )
3 equtr 1850 . . . . 5  |-  ( x  =  y  ->  (
y  =  z  ->  x  =  z )
)
43ancrd 556 . . . 4  |-  ( x  =  y  ->  (
y  =  z  -> 
( x  =  z  /\  y  =  z ) ) )
52, 4syl5 33 . . 3  |-  ( x  =  y  ->  (
z  =  y  -> 
( x  =  z  /\  y  =  z ) ) )
65eximdv 1758 . 2  |-  ( x  =  y  ->  ( E. z  z  =  y  ->  E. z ( x  =  z  /\  y  =  z ) ) )
71, 6mpi 20 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  y  =  z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   E.wex 1657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658
This theorem is referenced by:  equvin  1858  axc9lem1  2059  nfeqf  2104
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