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Theorem ax12vALT 2153
Description: Alternative proof of ax12v 1804, shorter, but depending on more axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax12vALT  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax12vALT
StepHypRef Expression
1 ax-1 6 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ph ) )
2 axc16 1888 . . . 4  |-  ( A. x  x  =  y  ->  ( ( x  =  y  ->  ph )  ->  A. x ( x  =  y  ->  ph ) ) )
31, 2syl5 32 . . 3  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x
( x  =  y  ->  ph ) ) )
43a1d 25 . 2  |-  ( A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
5 axc15 2058 . 2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
64, 5pm2.61i 164 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by: (None)
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