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Theorem a12study2 29756
Description: Reprove ax12o 1887 using dvelimhw 1747, showing that ax12o 1887 can be replaced by dveeq2 1893 (whose needed instances are the hypotheses here) if we allow distinct variables in axioms other than ax-17 1606. (Contributed by Andrew Salmon, 21-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
a12study2.1  |-  ( -. 
A. x  x  =  z  ->  ( w  =  z  ->  A. x  w  =  z )
)
a12study2.2  |-  ( -. 
A. x  x  =  y  ->  ( w  =  y  ->  A. x  w  =  y )
)
Assertion
Ref Expression
a12study2  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
Distinct variable groups:    x, w    y, w    z, w

Proof of Theorem a12study2
StepHypRef Expression
1 hbn1 1716 . . . . 5  |-  ( -. 
A. x  x  =  z  ->  A. x  -.  A. x  x  =  z )
2 a12study2.1 . . . . 5  |-  ( -. 
A. x  x  =  z  ->  ( w  =  z  ->  A. x  w  =  z )
)
31, 2hbim1 1744 . . . 4  |-  ( ( -.  A. x  x  =  z  ->  w  =  z )  ->  A. x ( -.  A. x  x  =  z  ->  w  =  z ) )
4 ax-17 1606 . . . 4  |-  ( ( -.  A. x  x  =  z  ->  y  =  z )  ->  A. w ( -.  A. x  x  =  z  ->  y  =  z ) )
5 equequ1 1667 . . . . 5  |-  ( w  =  y  ->  (
w  =  z  <->  y  =  z ) )
65imbi2d 307 . . . 4  |-  ( w  =  y  ->  (
( -.  A. x  x  =  z  ->  w  =  z )  <->  ( -.  A. x  x  =  z  ->  y  =  z ) ) )
73, 4, 6dvelimh 1917 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( ( -.  A. x  x  =  z  ->  y  =  z )  ->  A. x
( -.  A. x  x  =  z  ->  y  =  z ) ) )
8 nfa1 1768 . . . . 5  |-  F/ x A. x  x  =  z
98nfn 1777 . . . 4  |-  F/ x  -.  A. x  x  =  z
10919.21 1803 . . 3  |-  ( A. x ( -.  A. x  x  =  z  ->  y  =  z )  <-> 
( -.  A. x  x  =  z  ->  A. x  y  =  z ) )
117, 10syl6ib 217 . 2  |-  ( -. 
A. x  x  =  y  ->  ( ( -.  A. x  x  =  z  ->  y  =  z )  ->  ( -.  A. x  x  =  z  ->  A. x  y  =  z )
) )
1211pm2.86d 93 1  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530    = wceq 1632
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  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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