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Theorem ax12olem5 1872
Description: Lemma for ax12o 1875. See ax12olem6 1873 for derivation of ax12o 1875 from the conclusion. (Contributed by NM, 24-Dec-2015.)
Hypothesis
Ref Expression
ax12olem5.1  |-  ( -.  x  =  y  -> 
( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )
Assertion
Ref Expression
ax12olem5  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)

Proof of Theorem ax12olem5
StepHypRef Expression
1 exnal 1561 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2 19.8a 1718 . . 3  |-  ( y  =  z  ->  E. x  y  =  z )
3 hbe1 1705 . . . . 5  |-  ( E. x  y  =  z  ->  A. x E. x  y  =  z )
4 hba1 1719 . . . . 5  |-  ( A. x  y  =  z  ->  A. x A. x  y  =  z )
53, 4hbim 1725 . . . 4  |-  ( ( E. x  y  =  z  ->  A. x  y  =  z )  ->  A. x ( E. x  y  =  z  ->  A. x  y  =  z ) )
6 df-ex 1529 . . . . 5  |-  ( E. x  y  =  z  <->  -.  A. x  -.  y  =  z )
7 ax12olem5.1 . . . . 5  |-  ( -.  x  =  y  -> 
( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )
86, 7syl5bi 208 . . . 4  |-  ( -.  x  =  y  -> 
( E. x  y  =  z  ->  A. x  y  =  z )
)
95, 8exlimih 1729 . . 3  |-  ( E. x  -.  x  =  y  ->  ( E. x  y  =  z  ->  A. x  y  =  z ) )
102, 9syl5 28 . 2  |-  ( E. x  -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
111, 10sylbir 204 1  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527   E.wex 1528
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-11 1715
This theorem depends on definitions:  df-bi 177  df-ex 1529
  Copyright terms: Public domain W3C validator