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Theorem a12stdy1 29748
Description: Part of a study related to ax12o 1887. The consequent introduces a new variable  z. There are no distinct variable restrictions. (Contributed by NM, 14-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a12stdy1  |-  ( A. x  x  =  y  ->  E. x  y  =  z )

Proof of Theorem a12stdy1
StepHypRef Expression
1 a9e 1904 . 2  |-  E. y 
y  =  z
2 ax10o 1905 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x  -.  y  =  z  ->  A. y  -.  y  =  z ) )
32con3d 125 . . 3  |-  ( A. x  x  =  y  ->  ( -.  A. y  -.  y  =  z  ->  -.  A. x  -.  y  =  z )
)
4 df-ex 1532 . . 3  |-  ( E. y  y  =  z  <->  -.  A. y  -.  y  =  z )
5 df-ex 1532 . . 3  |-  ( E. x  y  =  z  <->  -.  A. x  -.  y  =  z )
63, 4, 53imtr4g 261 . 2  |-  ( A. x  x  =  y  ->  ( E. y  y  =  z  ->  E. x  y  =  z )
)
71, 6mpi 16 1  |-  ( A. x  x  =  y  ->  E. x  y  =  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   E.wex 1531
This theorem is referenced by:  a12stdy3  29750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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