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Theorem ax10lem18ALT 29124
Description: Distinctor with bound variable change without using sp 1716, ax9 1889, or ax10 1884 but allowing ax9v 1636. Uses ax12o 1875. (Contributed by NM, 22-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax10lem18ALT  |-  ( -. 
A. y  y  =  x  ->  ( A. x  x  =  w  ->  A. y  y  =  x ) )
Distinct variable groups:    x, w    y, w

Proof of Theorem ax10lem18ALT
StepHypRef Expression
1 sp 1716 . . 3  |-  ( A. x  x  =  w  ->  x  =  w )
2 ax10lem1 1876 . . . 4  |-  ( A. x  x  =  w  ->  A. y  y  =  w )
3 equcomi 1646 . . . . 5  |-  ( x  =  w  ->  w  =  x )
4 ax10lem17ALT 29123 . . . . . 6  |-  ( -. 
A. y  y  =  x  ->  ( w  =  x  ->  A. y  w  =  x )
)
5 equcomi 1646 . . . . . . 7  |-  ( w  =  x  ->  x  =  w )
65alimi 1546 . . . . . 6  |-  ( A. y  w  =  x  ->  A. y  x  =  w )
74, 6syl6 29 . . . . 5  |-  ( -. 
A. y  y  =  x  ->  ( w  =  x  ->  A. y  x  =  w )
)
8 hba1 1719 . . . . . . 7  |-  ( A. y  x  =  w  ->  A. y A. y  x  =  w )
9 sp 1716 . . . . . . . 8  |-  ( A. y  x  =  w  ->  x  =  w )
10 equequ2 1649 . . . . . . . 8  |-  ( x  =  w  ->  (
y  =  x  <->  y  =  w ) )
119, 10syl 15 . . . . . . 7  |-  ( A. y  x  =  w  ->  ( y  =  x  <-> 
y  =  w ) )
128, 11albidh 1577 . . . . . 6  |-  ( A. y  x  =  w  ->  ( A. y  y  =  x  <->  A. y 
y  =  w ) )
1312biimprd 214 . . . . 5  |-  ( A. y  x  =  w  ->  ( A. y  y  =  w  ->  A. y 
y  =  x ) )
143, 7, 13syl56 30 . . . 4  |-  ( -. 
A. y  y  =  x  ->  ( x  =  w  ->  ( A. y  y  =  w  ->  A. y  y  =  x ) ) )
152, 14syl7 63 . . 3  |-  ( -. 
A. y  y  =  x  ->  ( x  =  w  ->  ( A. x  x  =  w  ->  A. y  y  =  x ) ) )
161, 15syl5 28 . 2  |-  ( -. 
A. y  y  =  x  ->  ( A. x  x  =  w  ->  ( A. x  x  =  w  ->  A. y 
y  =  x ) ) )
1716pm2.43d 44 1  |-  ( -. 
A. y  y  =  x  ->  ( A. x  x  =  w  ->  A. y  y  =  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   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  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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