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Theorem ax12vALT 2131
Description: Alternate proof of ax12v 2130 that avoids theorem axc16 2011 and is proved directly from ax-12 1792 rather than via axc15 2035. (Contributed by Jim Kingdon, 15-Dec-2017.) (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
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1710 . 2  |-  E. z 
z  =  y
2 ax-5 1670 . . . . 5  |-  ( ph  ->  A. z ph )
3 ax-12 1792 . . . . 5  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
42, 3syl5 32 . . . 4  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
5 equequ2 1737 . . . . 5  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
65imbi1d 317 . . . . . . 7  |-  ( z  =  y  ->  (
( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
76albidv 1679 . . . . . 6  |-  ( z  =  y  ->  ( A. x ( x  =  z  ->  ph )  <->  A. x
( x  =  y  ->  ph ) ) )
87imbi2d 316 . . . . 5  |-  ( z  =  y  ->  (
( ph  ->  A. x
( x  =  z  ->  ph ) )  <->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
95, 8imbi12d 320 . . . 4  |-  ( z  =  y  ->  (
( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph ) ) )  <->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
104, 9mpbii 211 . . 3  |-  ( z  =  y  ->  (
x  =  y  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
1110exlimiv 1688 . 2  |-  ( E. z  z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
121, 11ax-mp 5 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1367    = wceq 1369   E.wex 1586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-ex 1587
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator