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Theorem ax12v 1945
Description: This is a version of ax-12 1944 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax12v2 2186 for the rederivation of ax-c15 32506 from this theorem. See ax12vALT 2268 for a shorter proof requiring more axioms. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1926 and ax-13 2102. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
Assertion
Ref Expression
ax12v  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax12v
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 equequ2 1879 . . . 4  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
21biimprd 231 . . 3  |-  ( z  =  y  ->  (
x  =  y  ->  x  =  z )
)
3 ax-5 1769 . . . . 5  |-  ( ph  ->  A. z ph )
4 ax-12 1944 . . . . 5  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
53, 4syl5 33 . . . 4  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
62imim1d 78 . . . . 5  |-  ( z  =  y  ->  (
( x  =  z  ->  ph )  ->  (
x  =  y  ->  ph ) ) )
76alimdv 1774 . . . 4  |-  ( z  =  y  ->  ( A. x ( x  =  z  ->  ph )  ->  A. x ( x  =  y  ->  ph ) ) )
85, 7syl9r 74 . . 3  |-  ( z  =  y  ->  (
x  =  z  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
92, 8syld 45 . 2  |-  ( z  =  y  ->  (
x  =  y  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
10 ax6ev 1818 . 2  |-  E. z 
z  =  y
119, 10exlimiiv 1788 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-12 1944
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675
This theorem is referenced by:  19.8a  1946  19.8aOLD  1947  sb56  2092  ax12a2  2187  exsb  2308  mo2v  2317  2eu6  2398  bj-ax12  31292  bj-ssbequ1  31302  bj-ssbid1ALT  31306  wl-lem-exsb  31940  wl-lem-moexsb  31942  rexsb  38627
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