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Theorem ax12olem6 1873
Description: Lemma for ax12o 1875. Derivation of ax12o 1875 from the hypotheses, without using ax12o 1875. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.)
Hypotheses
Ref Expression
ax12olem6.1  |-  ( -. 
A. x  x  =  z  ->  ( z  =  w  ->  A. x  z  =  w )
)
ax12olem6.2  |-  ( -. 
A. x  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w )
)
Assertion
Ref Expression
ax12olem6  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
Distinct variable groups:    x, w    y, w    z, w

Proof of Theorem ax12olem6
StepHypRef Expression
1 hbn1 1704 . . . . 5  |-  ( -. 
A. x  x  =  z  ->  A. x  -.  A. x  x  =  z )
2 ax12olem6.1 . . . . 5  |-  ( -. 
A. x  x  =  z  ->  ( z  =  w  ->  A. x  z  =  w )
)
31, 2hbim1 1732 . . . 4  |-  ( ( -.  A. x  x  =  z  ->  z  =  w )  ->  A. x
( -.  A. x  x  =  z  ->  z  =  w ) )
4 ax-17 1603 . . . 4  |-  ( ( -.  A. x  x  =  z  ->  y  =  z )  ->  A. w ( -.  A. x  x  =  z  ->  y  =  z ) )
5 equcom 1647 . . . . . 6  |-  ( z  =  w  <->  w  =  z )
6 equequ1 1648 . . . . . 6  |-  ( w  =  y  ->  (
w  =  z  <->  y  =  z ) )
75, 6syl5bb 248 . . . . 5  |-  ( w  =  y  ->  (
z  =  w  <->  y  =  z ) )
87imbi2d 307 . . . 4  |-  ( w  =  y  ->  (
( -.  A. x  x  =  z  ->  z  =  w )  <->  ( -.  A. x  x  =  z  ->  y  =  z ) ) )
9 ax12olem6.2 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w )
)
103, 4, 8, 9dvelimhw 1735 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( ( -.  A. x  x  =  z  ->  y  =  z )  ->  A. x
( -.  A. x  x  =  z  ->  y  =  z ) ) )
11119.21h 1731 . . 3  |-  ( A. x ( -.  A. x  x  =  z  ->  y  =  z )  <-> 
( -.  A. x  x  =  z  ->  A. x  y  =  z ) )
1210, 11syl6ib 217 . 2  |-  ( -. 
A. x  x  =  y  ->  ( ( -.  A. x  x  =  z  ->  y  =  z )  ->  ( -.  A. x  x  =  z  ->  A. x  y  =  z )
) )
1312pm2.86d 93 1  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem is referenced by:  ax12olem7  1874
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
This theorem depends on definitions:  df-bi 177
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