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Theorem ax12 1882
Description: Derive ax-12 1633 from ax-12o 1664. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)
Assertion
Ref Expression
ax12  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )

Proof of Theorem ax12
StepHypRef Expression
1 ax-4 1692 . . . . . 6  |-  ( A. x  x  =  y  ->  x  =  y )
21con3i 129 . . . . 5  |-  ( -.  x  =  y  ->  -.  A. x  x  =  y )
32adantr 453 . . . 4  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  A. x  x  =  y )
4 equtrr 1827 . . . . . . . 8  |-  ( z  =  y  ->  (
x  =  z  ->  x  =  y )
)
54equcoms 1825 . . . . . . 7  |-  ( y  =  z  ->  (
x  =  z  ->  x  =  y )
)
65con3rr3 130 . . . . . 6  |-  ( -.  x  =  y  -> 
( y  =  z  ->  -.  x  =  z ) )
76imp 420 . . . . 5  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  x  =  z )
8 ax-4 1692 . . . . 5  |-  ( A. x  x  =  z  ->  x  =  z )
97, 8nsyl 115 . . . 4  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  A. x  x  =  z )
10 ax-12o 1664 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
113, 9, 10sylc 58 . . 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 46 1  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   A.wal 1532
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-8 1623  ax-17 1628  ax-12o 1664  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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