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Theorem ax12-3 29104
Description: An equivalent to ax12-2 29103. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax12-3  |-  ( E. z  -.  x  =  y  ->  ( E. z  x  =  y  ->  E. x  z  =  y ) )

Proof of Theorem ax12-3
StepHypRef Expression
1 exnal 1561 . 2  |-  ( E. z  -.  x  =  y  <->  -.  A. z  x  =  y )
2 ax12-2 29103 . . . . . 6  |-  ( A. x  -.  z  =  y  ->  ( -.  A. z  -.  x  =  y  ->  A. z  x  =  y ) )
32con1d 116 . . . . 5  |-  ( A. x  -.  z  =  y  ->  ( -.  A. z  x  =  y  ->  A. z  -.  x  =  y ) )
43com12 27 . . . 4  |-  ( -. 
A. z  x  =  y  ->  ( A. x  -.  z  =  y  ->  A. z  -.  x  =  y ) )
54con3d 125 . . 3  |-  ( -. 
A. z  x  =  y  ->  ( -.  A. z  -.  x  =  y  ->  -.  A. x  -.  z  =  y
) )
6 df-ex 1529 . . 3  |-  ( E. z  x  =  y  <->  -.  A. z  -.  x  =  y )
7 df-ex 1529 . . 3  |-  ( E. x  z  =  y  <->  -.  A. x  -.  z  =  y )
85, 6, 73imtr4g 261 . 2  |-  ( -. 
A. z  x  =  y  ->  ( E. z  x  =  y  ->  E. x  z  =  y ) )
91, 8sylbi 187 1  |-  ( E. z  -.  x  =  y  ->  ( E. z  x  =  y  ->  E. x  z  =  y ) )
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-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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