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Theorem a12stdy1-x12 29111
Description: Part of a study related to ax12o 1875. Weak version of a12stdy1 29126. Does not use sp 1716, ax9 1889, ax10 1884, or ax12o 1875 but allows ax9v 1636. The consequent introduces a new variable  z. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a12stdy1-x12  |-  ( A. x  x  =  y  ->  E. x  y  =  z )
Distinct variable groups:    x, y    y, z

Proof of Theorem a12stdy1-x12
StepHypRef Expression
1 ax9v 1636 . . 3  |-  -.  A. y  -.  y  =  z
2 df-ex 1529 . . 3  |-  ( E. y  y  =  z  <->  -.  A. y  -.  y  =  z )
31, 2mpbir 200 . 2  |-  E. y 
y  =  z
4 ax10lem3 1878 . . . . 5  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
5 ax10lem6 1883 . . . . 5  |-  ( A. y  y  =  x  ->  ( A. x  -.  y  =  z  ->  A. y  -.  y  =  z ) )
64, 5syl 15 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x  -.  y  =  z  ->  A. y  -.  y  =  z ) )
76con3d 125 . . 3  |-  ( A. x  x  =  y  ->  ( -.  A. y  -.  y  =  z  ->  -.  A. x  -.  y  =  z )
)
8 df-ex 1529 . . 3  |-  ( E. x  y  =  z  <->  -.  A. x  -.  y  =  z )
97, 2, 83imtr4g 261 . 2  |-  ( A. x  x  =  y  ->  ( E. y  y  =  z  ->  E. x  y  =  z )
)
103, 9mpi 16 1  |-  ( A. x  x  =  y  ->  E. x  y  =  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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