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Theorem ax12OLD 2021
Description: Obsolete proof of ax12 2020 as of 31-Jan-2018. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax12OLD  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )

Proof of Theorem ax12OLD
StepHypRef Expression
1 sp 1764 . . . . . 6  |-  ( A. x  x  =  y  ->  x  =  y )
21con3i 130 . . . . 5  |-  ( -.  x  =  y  ->  -.  A. x  x  =  y )
32adantr 453 . . . 4  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  A. x  x  =  y )
4 equtrr 1696 . . . . . . . 8  |-  ( z  =  y  ->  (
x  =  z  ->  x  =  y )
)
54equcoms 1694 . . . . . . 7  |-  ( y  =  z  ->  (
x  =  z  ->  x  =  y )
)
65con3rr3 131 . . . . . 6  |-  ( -.  x  =  y  -> 
( y  =  z  ->  -.  x  =  z ) )
76imp 420 . . . . 5  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  x  =  z )
8 sp 1764 . . . . 5  |-  ( A. x  x  =  z  ->  x  =  z )
97, 8nsyl 116 . . . 4  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  A. x  x  =  z )
10 ax12o 2011 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
113, 9, 10sylc 59 . . 3  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  ( y  =  z  ->  A. x  y  =  z )
)
1211ex 425 . 2  |-  ( -.  x  =  y  -> 
( y  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
1312pm2.43d 47 1  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wal 1550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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