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Theorem ax12olem7 1874
Description: Lemma for ax12o 1875. Derivation of ax12o 1875 from the hypotheses, without using ax12o 1875. (Contributed by NM, 24-Dec-2015.)
Hypotheses
Ref Expression
ax12olem7.1  |-  ( -.  x  =  z  -> 
( -.  A. x  -.  z  =  w  ->  A. x  z  =  w ) )
ax12olem7.2  |-  ( -.  x  =  y  -> 
( -.  A. x  -.  y  =  w  ->  A. x  y  =  w ) )
Assertion
Ref Expression
ax12olem7  |-  ( -. 
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 ax12olem7
StepHypRef Expression
1 ax12olem7.1 . . 3  |-  ( -.  x  =  z  -> 
( -.  A. x  -.  z  =  w  ->  A. x  z  =  w ) )
21ax12olem5 1872 . 2  |-  ( -. 
A. x  x  =  z  ->  ( z  =  w  ->  A. x  z  =  w )
)
3 ax12olem7.2 . . 3  |-  ( -.  x  =  y  -> 
( -.  A. x  -.  y  =  w  ->  A. x  y  =  w ) )
43ax12olem5 1872 . 2  |-  ( -. 
A. x  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w )
)
52, 4ax12olem6 1873 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:  ax12o  1875
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  df-ex 1529
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