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Theorem a12study2 29134
Description: Reprove ax12o 1875 using dvelimhw 1735, showing that ax12o 1875 can be replaced by dveeq2 1880 (whose needed instances are the hypotheses here) if we allow distinct variables in axioms other than ax-17 1603. (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 1704 . . . . 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 1732 . . . 4  |-  ( ( -.  A. x  x  =  z  ->  w  =  z )  ->  A. x ( -.  A. x  x  =  z  ->  w  =  z ) )
4 ax-17 1603 . . . 4  |-  ( ( -.  A. x  x  =  z  ->  y  =  z )  ->  A. w ( -.  A. x  x  =  z  ->  y  =  z ) )
5 equequ1 1648 . . . . 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 1904 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( ( -.  A. x  x  =  z  ->  y  =  z )  ->  A. x
( -.  A. x  x  =  z  ->  y  =  z ) ) )
8 nfa1 1756 . . . . 5  |-  F/ x A. x  x  =  z
98nfn 1765 . . . 4  |-  F/ x  -.  A. x  x  =  z
10919.21 1791 . . 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 1527    = wceq 1623
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  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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