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Theorem ax12inda 2249
Description: Induction step for constructing a substitution instance of ax-c15 2191 without using ax-c15 2191. Quantification case. (When  z and  y are distinct, ax12inda2 2248 may be used instead to avoid the dummy variable  w in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax12inda.1  |-  ( -. 
A. x  x  =  w  ->  ( x  =  w  ->  ( ph  ->  A. x ( x  =  w  ->  ph )
) ) )
Assertion
Ref Expression
ax12inda  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) )
Distinct variable groups:    ph, w    x, w    y, w    z, w
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem ax12inda
StepHypRef Expression
1 ax6ev 1710 . . 3  |-  E. w  w  =  y
2 ax12inda.1 . . . . . . 7  |-  ( -. 
A. x  x  =  w  ->  ( x  =  w  ->  ( ph  ->  A. x ( x  =  w  ->  ph )
) ) )
32ax12inda2 2248 . . . . . 6  |-  ( -. 
A. x  x  =  w  ->  ( x  =  w  ->  ( A. z ph  ->  A. x
( x  =  w  ->  A. z ph )
) ) )
4 dveeq2-o 2234 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( w  =  y  ->  A. x  w  =  y )
)
54imp 429 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  ->  A. x  w  =  y )
6 hba1-o 2199 . . . . . . . . . 10  |-  ( A. x  w  =  y  ->  A. x A. x  w  =  y )
7 equequ2 1737 . . . . . . . . . . 11  |-  ( w  =  y  ->  (
x  =  w  <->  x  =  y ) )
87sps-o 2209 . . . . . . . . . 10  |-  ( A. x  w  =  y  ->  ( x  =  w  <-> 
x  =  y ) )
96, 8albidh 1642 . . . . . . . . 9  |-  ( A. x  w  =  y  ->  ( A. x  x  =  w  <->  A. x  x  =  y )
)
109notbid 294 . . . . . . . 8  |-  ( A. x  w  =  y  ->  ( -.  A. x  x  =  w  <->  -.  A. x  x  =  y )
)
115, 10syl 16 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( -.  A. x  x  =  w  <->  -.  A. x  x  =  y )
)
127adantl 466 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( x  =  w  <-> 
x  =  y ) )
138imbi1d 317 . . . . . . . . . . 11  |-  ( A. x  w  =  y  ->  ( ( x  =  w  ->  A. z ph )  <->  ( x  =  y  ->  A. z ph ) ) )
146, 13albidh 1642 . . . . . . . . . 10  |-  ( A. x  w  =  y  ->  ( A. x ( x  =  w  ->  A. z ph )  <->  A. x
( x  =  y  ->  A. z ph )
) )
155, 14syl 16 . . . . . . . . 9  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( A. x ( x  =  w  ->  A. z ph )  <->  A. x
( x  =  y  ->  A. z ph )
) )
1615imbi2d 316 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( ( A. z ph  ->  A. x ( x  =  w  ->  A. z ph ) )  <->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) )
1712, 16imbi12d 320 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( ( x  =  w  ->  ( A. z ph  ->  A. x
( x  =  w  ->  A. z ph )
) )  <->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) ) )
1811, 17imbi12d 320 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( ( -.  A. x  x  =  w  ->  ( x  =  w  ->  ( A. z ph  ->  A. x ( x  =  w  ->  A. z ph ) ) ) )  <-> 
( -.  A. x  x  =  y  ->  ( x  =  y  -> 
( A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) ) ) )
193, 18mpbii 211 . . . . 5  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( -.  A. x  x  =  y  ->  ( x  =  y  -> 
( A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) ) )
2019ex 434 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( w  =  y  ->  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) ) ) )
2120exlimdv 1690 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( E. w  w  =  y  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  -> 
( A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) ) ) )
221, 21mpi 17 . 2  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) ) )
2322pm2.43i 47 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   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-11 1780  ax-12 1792  ax-13 1943  ax-c5 2185  ax-c4 2186  ax-c7 2187  ax-c11 2189  ax-c9 2192  ax-c16 2194
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590
This theorem is referenced by: (None)
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