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Theorem ax12b 1696
Description: Two equivalent ways of expressing ax-12 1939. See the comment for ax-12 1939. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2017.)
Assertion
Ref Expression
ax12b  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  <->  ( -.  x  =  y  ->  ( -.  x  =  z  -> 
( y  =  z  ->  A. x  y  =  z ) ) ) )

Proof of Theorem ax12b
StepHypRef Expression
1 id 20 . . 3  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  ->  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) ) )
21a1dd 44 . 2  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  ->  ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) ) )
3 equtrr 1690 . . . . . 6  |-  ( z  =  y  ->  (
x  =  z  ->  x  =  y )
)
43equcoms 1688 . . . . 5  |-  ( y  =  z  ->  (
x  =  z  ->  x  =  y )
)
54con3rr3 130 . . . 4  |-  ( -.  x  =  y  -> 
( y  =  z  ->  -.  x  =  z ) )
6 id 20 . . . . . 6  |-  ( ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )  ->  ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) ) )
76com4l 80 . . . . 5  |-  ( -.  x  =  y  -> 
( -.  x  =  z  ->  ( y  =  z  ->  ( ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )  ->  A. x  y  =  z )
) ) )
87com23 74 . . . 4  |-  ( -.  x  =  y  -> 
( y  =  z  ->  ( -.  x  =  z  ->  ( ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )  ->  A. x  y  =  z )
) ) )
95, 8mpdd 38 . . 3  |-  ( -.  x  =  y  -> 
( y  =  z  ->  ( ( -.  x  =  y  -> 
( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )  ->  A. x  y  =  z )
) )
109com3r 75 . 2  |-  ( ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )  ->  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) ) )
112, 10impbii 181 1  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  <->  ( -.  x  =  y  ->  ( -.  x  =  z  -> 
( y  =  z  ->  A. x  y  =  z ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1546
This theorem is referenced by:  ax12  1975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682
This theorem depends on definitions:  df-bi 178  df-ex 1548
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