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Theorem ax12b 1701
Description: Two equivalent ways of expressing ax-12 1950. See the comment for ax-12 1950. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.)
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 ax-1 5 . . 3  |-  ( ( y  =  z  ->  A. x  y  =  z )  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) )
2 equtrr 1695 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  ->  x  =  y )
)
32equcoms 1693 . . . . . 6  |-  ( y  =  z  ->  (
x  =  z  ->  x  =  y )
)
43con3rr3 130 . . . . 5  |-  ( -.  x  =  y  -> 
( y  =  z  ->  -.  x  =  z ) )
54imim1d 71 . . . 4  |-  ( -.  x  =  y  -> 
( ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
)  ->  ( y  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) ) )
6 pm2.43 49 . . . 4  |-  ( ( y  =  z  -> 
( y  =  z  ->  A. x  y  =  z ) )  -> 
( y  =  z  ->  A. x  y  =  z ) )
75, 6syl6 31 . . 3  |-  ( -.  x  =  y  -> 
( ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
)  ->  ( y  =  z  ->  A. x  y  =  z )
) )
81, 7impbid2 196 . 2  |-  ( -.  x  =  y  -> 
( ( y  =  z  ->  A. x  y  =  z )  <->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) ) )
98pm5.74i 237 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 1549
This theorem is referenced by:  ax12  2019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687
This theorem depends on definitions:  df-bi 178  df-ex 1551
  Copyright terms: Public domain W3C validator