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Theorem wl-ax12 31877
Description: Rederivation of axiom ax-12 1909 from ax12v 1910, axc112 1997, and other axioms. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.)
Assertion
Ref Expression
wl-ax12  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )

Proof of Theorem wl-ax12
StepHypRef Expression
1 biidd 240 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ph ) )
21dral1 2126 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ph ) )
3 ax-1 6 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ph ) )
43alimi 1678 . . . 4  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
52, 4syl6bir 232 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) )
65a1d 26 . 2  |-  ( A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) ) )
7 sp 1914 . . 3  |-  ( A. y ph  ->  ph )
8 wl-ax12v3 31876 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
97, 8syl7 70 . 2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
106, 9pm2.61i 167 1  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435
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  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by: (None)
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