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Theorem ax12bOLD 1656
Description: Obsolete version of ax12b 1655 as of 12-Aug-2017. (Contributed by NM, 2-May-2017.) (New usage is discouraged.)
Assertion
Ref Expression
ax12bOLD  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  <->  ( -.  x  =  y  ->  ( -.  x  =  z  -> 
( y  =  z  ->  A. x  y  =  z ) ) ) )

Proof of Theorem ax12bOLD
StepHypRef Expression
1 bi2.04 350 . . . 4  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  <->  ( y  =  z  ->  ( -.  x  =  y  ->  A. x  y  =  z ) ) )
2 equtrr 1653 . . . . . . . . 9  |-  ( z  =  y  ->  (
x  =  z  ->  x  =  y )
)
32equcoms 1651 . . . . . . . 8  |-  ( y  =  z  ->  (
x  =  z  ->  x  =  y )
)
43con3d 125 . . . . . . 7  |-  ( y  =  z  ->  ( -.  x  =  y  ->  -.  x  =  z ) )
54pm4.71d 615 . . . . . 6  |-  ( y  =  z  ->  ( -.  x  =  y  <->  ( -.  x  =  y  /\  -.  x  =  z ) ) )
65imbi1d 308 . . . . 5  |-  ( y  =  z  ->  (
( -.  x  =  y  ->  A. x  y  =  z )  <->  ( ( -.  x  =  y  /\  -.  x  =  z )  ->  A. x  y  =  z ) ) )
76pm5.74i 236 . . . 4  |-  ( ( y  =  z  -> 
( -.  x  =  y  ->  A. x  y  =  z )
)  <->  ( y  =  z  ->  ( ( -.  x  =  y  /\  -.  x  =  z )  ->  A. x  y  =  z )
) )
81, 7bitri 240 . . 3  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  <->  ( y  =  z  ->  ( ( -.  x  =  y  /\  -.  x  =  z )  ->  A. x  y  =  z )
) )
9 bi2.04 350 . . 3  |-  ( ( y  =  z  -> 
( ( -.  x  =  y  /\  -.  x  =  z )  ->  A. x  y  =  z ) )  <->  ( ( -.  x  =  y  /\  -.  x  =  z )  ->  ( y  =  z  ->  A. x  y  =  z )
) )
108, 9bitri 240 . 2  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  <->  ( ( -.  x  =  y  /\  -.  x  =  z
)  ->  ( y  =  z  ->  A. x  y  =  z )
) )
11 impexp 433 . 2  |-  ( ( ( -.  x  =  y  /\  -.  x  =  z )  -> 
( y  =  z  ->  A. x  y  =  z ) )  <->  ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) ) )
1210, 11bitri 240 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 176    /\ wa 358   A.wal 1527
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
This theorem depends on definitions:  df-bi 177  df-an 360
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