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Theorem bj-axc15v 32361
Description: Version of axc15 2035 with a dv condition, which does not require ax-13 1943. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axc15v  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem bj-axc15v
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1710 . 2  |-  E. z 
z  =  y
2 ax5d 1671 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
3 ax-5 1670 . . . . . 6  |-  ( ph  ->  A. z ph )
4 ax-12 1792 . . . . . 6  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
53, 4syl5 32 . . . . 5  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
6 equequ2 1737 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
76sps 1800 . . . . . 6  |-  ( A. x  z  =  y  ->  ( x  =  z  <-> 
x  =  y ) )
8 nfa1 1831 . . . . . . . 8  |-  F/ x A. x  z  =  y
97imbi1d 317 . . . . . . . 8  |-  ( A. x  z  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
108, 9albid 1819 . . . . . . 7  |-  ( A. x  z  =  y  ->  ( A. x ( x  =  z  ->  ph )  <->  A. x ( x  =  y  ->  ph )
) )
1110imbi2d 316 . . . . . 6  |-  ( A. x  z  =  y  ->  ( ( ph  ->  A. x ( x  =  z  ->  ph ) )  <-> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
127, 11imbi12d 320 . . . . 5  |-  ( A. x  z  =  y  ->  ( ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )  <->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
135, 12mpbii 211 . . . 4  |-  ( A. x  z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
142, 13syl6 33 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
1514exlimdv 1690 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. z  z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) ) )
161, 15mpi 17 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1367   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-10 1775  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-ex 1587  df-nf 1590
This theorem is referenced by:  bj-equs5v  32363  bj-ax12v  32379
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