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Theorem ax13b 1859
Description: An equivalence used to show two ways of expressing ax-13 2057. See the comment for ax-13 2057. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)
Assertion
Ref Expression
ax13b  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  ph ) )  <-> 
( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  ph ) ) ) )

Proof of Theorem ax13b
StepHypRef Expression
1 ax-1 6 . . 3  |-  ( ( y  =  z  ->  ph )  ->  ( -.  x  =  z  -> 
( y  =  z  ->  ph ) ) )
2 equtrr 1851 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  ->  x  =  y )
)
32equcoms 1849 . . . . . 6  |-  ( y  =  z  ->  (
x  =  z  ->  x  =  y )
)
43con3rr3 141 . . . . 5  |-  ( -.  x  =  y  -> 
( y  =  z  ->  -.  x  =  z ) )
54imim1d 78 . . . 4  |-  ( -.  x  =  y  -> 
( ( -.  x  =  z  ->  ( y  =  z  ->  ph )
)  ->  ( y  =  z  ->  ( y  =  z  ->  ph )
) ) )
6 pm2.43 53 . . . 4  |-  ( ( y  =  z  -> 
( y  =  z  ->  ph ) )  -> 
( y  =  z  ->  ph ) )
75, 6syl6 34 . . 3  |-  ( -.  x  =  y  -> 
( ( -.  x  =  z  ->  ( y  =  z  ->  ph )
)  ->  ( y  =  z  ->  ph )
) )
81, 7impbid2 207 . 2  |-  ( -.  x  =  y  -> 
( ( y  =  z  ->  ph )  <->  ( -.  x  =  z  ->  ( y  =  z  ->  ph ) ) ) )
98pm5.74i 248 1  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  ph ) )  <-> 
( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  ph ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843
This theorem depends on definitions:  df-bi 188  df-ex 1658
This theorem is referenced by:  ax13  2106
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