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Theorem a12stdy4 29129
Description: Part of a study related to ax12o 1875. The second antecedent of ax12o 1875 is replaced. There are no distinct variable restrictions. (Contributed by NM, 14-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a12stdy4  |-  ( -. 
A. z  z  =  x  ->  ( A. y  z  =  x  ->  ( x  =  y  ->  A. z  x  =  y ) ) )

Proof of Theorem a12stdy4
StepHypRef Expression
1 ax10o 1892 . . . . . . 7  |-  ( A. y  y  =  z  ->  ( A. y  z  =  x  ->  A. z 
z  =  x ) )
21aecoms 1887 . . . . . 6  |-  ( A. z  z  =  y  ->  ( A. y  z  =  x  ->  A. z 
z  =  x ) )
32con3d 125 . . . . 5  |-  ( A. z  z  =  y  ->  ( -.  A. z 
z  =  x  ->  -.  A. y  z  =  x ) )
43impcom 419 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  A. z  z  =  y
)  ->  -.  A. y 
z  =  x )
54pm2.21d 98 . . 3  |-  ( ( -.  A. z  z  =  x  /\  A. z  z  =  y
)  ->  ( A. y  z  =  x  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
65ex 423 . 2  |-  ( -. 
A. z  z  =  x  ->  ( A. z  z  =  y  ->  ( A. y  z  =  x  ->  (
x  =  y  ->  A. z  x  =  y ) ) ) )
7 ax12o 1875 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
87a1dd 42 . 2  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( A. y 
z  =  x  -> 
( x  =  y  ->  A. z  x  =  y ) ) ) )
96, 8pm2.61d 150 1  |-  ( -. 
A. z  z  =  x  ->  ( A. y  z  =  x  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527
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
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