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Theorem ax12 1888
Description: Derive ax-12 1878 from ax12v 1879 via ax12o 1887. This shows that the weakening in ax12v 1879 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.)
Assertion
Ref Expression
ax12  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )

Proof of Theorem ax12
StepHypRef Expression
1 sp 1728 . . . . . 6  |-  ( A. x  x  =  y  ->  x  =  y )
21con3i 127 . . . . 5  |-  ( -.  x  =  y  ->  -.  A. x  x  =  y )
32adantr 451 . . . 4  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  A. x  x  =  y )
4 equtrr 1672 . . . . . . . 8  |-  ( z  =  y  ->  (
x  =  z  ->  x  =  y )
)
54equcoms 1666 . . . . . . 7  |-  ( y  =  z  ->  (
x  =  z  ->  x  =  y )
)
65con3rr3 128 . . . . . 6  |-  ( -.  x  =  y  -> 
( y  =  z  ->  -.  x  =  z ) )
76imp 418 . . . . 5  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  x  =  z )
8 sp 1728 . . . . 5  |-  ( A. x  x  =  z  ->  x  =  z )
97, 8nsyl 113 . . . 4  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  A. x  x  =  z )
10 ax12o 1887 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
113, 9, 10sylc 56 . . 3  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  ( y  =  z  ->  A. x  y  =  z )
)
1211ex 423 . 2  |-  ( -.  x  =  y  -> 
( y  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
1312pm2.43d 44 1  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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