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Theorem ax12v2 1952
Description: It is possible to remove any restriction on  ph in ax12v 1951. Same as Axiom C8 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1932 and ax-13 2104. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
Assertion
Ref Expression
ax12v2  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax12v2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 equtrr 1874 . . 3  |-  ( y  =  z  ->  (
x  =  y  ->  x  =  z )
)
2 ax12v 1951 . . . 4  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
31imim1d 77 . . . . 5  |-  ( y  =  z  ->  (
( x  =  z  ->  ph )  ->  (
x  =  y  ->  ph ) ) )
43alimdv 1771 . . . 4  |-  ( y  =  z  ->  ( A. x ( x  =  z  ->  ph )  ->  A. x ( x  =  y  ->  ph ) ) )
52, 4syl9r 73 . . 3  |-  ( y  =  z  ->  (
x  =  z  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
61, 5syld 44 . 2  |-  ( y  =  z  ->  (
x  =  y  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
7 ax6evr 1867 . 2  |-  E. z 
y  =  z
86, 7exlimiiv 1785 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672
This theorem is referenced by:  sb56  2096  bj-ax12  31311  wl-lem-exsb  31965  wl-lem-moexsb  31967
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