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Theorem ax12olem2 2006
Description: Lemma for nfeqf 2009 and dveeq1 2021. This lemma is equivalent to ax12v 1951 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 29-Apr-2018.)
Assertion
Ref Expression
ax12olem2  |-  ( -.  x  =  y  -> 
( E. x  y  =  z  ->  y  =  z ) )
Distinct variable group:    x, z

Proof of Theorem ax12olem2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax12v 1951 . . . 4  |-  ( -.  x  =  y  -> 
( y  =  w  ->  A. x  y  =  w ) )
2 ax-8 1687 . . . . . 6  |-  ( y  =  z  ->  (
y  =  w  -> 
z  =  w ) )
32eximi 1585 . . . . 5  |-  ( E. x  y  =  z  ->  E. x ( y  =  w  ->  z  =  w ) )
4 19.36v 1919 . . . . 5  |-  ( E. x ( y  =  w  ->  z  =  w )  <->  ( A. x  y  =  w  ->  z  =  w ) )
53, 4sylib 189 . . . 4  |-  ( E. x  y  =  z  ->  ( A. x  y  =  w  ->  z  =  w ) )
61, 5syl9 68 . . 3  |-  ( -.  x  =  y  -> 
( E. x  y  =  z  ->  (
y  =  w  -> 
z  =  w ) ) )
76alrimdv 1643 . 2  |-  ( -.  x  =  y  -> 
( E. x  y  =  z  ->  A. w
( y  =  w  ->  z  =  w ) ) )
8 ax12olem1 2005 . 2  |-  ( y  =  z  <->  A. w
( y  =  w  ->  z  =  w ) )
97, 8syl6ibr 219 1  |-  ( -.  x  =  y  -> 
( E. x  y  =  z  ->  y  =  z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549   E.wex 1550
This theorem is referenced by:  ax12olem3  2007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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