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Theorem ax12olem2 1881
Description: Lemma for ax12o 1887. Negate the equalities in ax-12 1878, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.)
Hypothesis
Ref Expression
ax12olem2.1  |-  ( -.  x  =  y  -> 
( y  =  w  ->  A. x  y  =  w ) )
Assertion
Ref Expression
ax12olem2  |-  ( -.  x  =  y  -> 
( -.  y  =  z  ->  A. x  -.  y  =  z
) )
Distinct variable groups:    x, w, z    y, w

Proof of Theorem ax12olem2
StepHypRef Expression
1 ax12olem2.1 . . . . . 6  |-  ( -.  x  =  y  -> 
( y  =  w  ->  A. x  y  =  w ) )
21anim1d 547 . . . . 5  |-  ( -.  x  =  y  -> 
( ( y  =  w  /\  -.  z  =  w )  ->  ( A. x  y  =  w  /\  -.  z  =  w ) ) )
3 ax-17 1606 . . . . . . 7  |-  ( -.  z  =  w  ->  A. x  -.  z  =  w )
43anim2i 552 . . . . . 6  |-  ( ( A. x  y  =  w  /\  -.  z  =  w )  ->  ( A. x  y  =  w  /\  A. x  -.  z  =  w )
)
5 19.26 1583 . . . . . 6  |-  ( A. x ( y  =  w  /\  -.  z  =  w )  <->  ( A. x  y  =  w  /\  A. x  -.  z  =  w ) )
64, 5sylibr 203 . . . . 5  |-  ( ( A. x  y  =  w  /\  -.  z  =  w )  ->  A. x
( y  =  w  /\  -.  z  =  w ) )
72, 6syl6 29 . . . 4  |-  ( -.  x  =  y  -> 
( ( y  =  w  /\  -.  z  =  w )  ->  A. x
( y  =  w  /\  -.  z  =  w ) ) )
87eximdv 1612 . . 3  |-  ( -.  x  =  y  -> 
( E. w ( y  =  w  /\  -.  z  =  w
)  ->  E. w A. x ( y  =  w  /\  -.  z  =  w ) ) )
9 19.12 1746 . . 3  |-  ( E. w A. x ( y  =  w  /\  -.  z  =  w
)  ->  A. x E. w ( y  =  w  /\  -.  z  =  w ) )
108, 9syl6 29 . 2  |-  ( -.  x  =  y  -> 
( E. w ( y  =  w  /\  -.  z  =  w
)  ->  A. x E. w ( y  =  w  /\  -.  z  =  w ) ) )
11 ax12olem1 1880 . 2  |-  ( E. w ( y  =  w  /\  -.  z  =  w )  <->  -.  y  =  z )
1211albii 1556 . 2  |-  ( A. x E. w ( y  =  w  /\  -.  z  =  w )  <->  A. x  -.  y  =  z )
1310, 11, 123imtr3g 260 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   E.wex 1531
This theorem is referenced by:  ax12olem4  1883
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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