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Theorem ax11inda 2152
Description: Induction step for constructing a substitution instance of ax-11o 2093 without using ax-11o 2093. Quantification case. (When  z and  y are distinct, ax11inda2 2151 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
ax11inda.1  |-  ( -. 
A. x  x  =  w  ->  ( x  =  w  ->  ( ph  ->  A. x ( x  =  w  ->  ph )
) ) )
Assertion
Ref Expression
ax11inda  |-  ( -. 
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 ax11inda
StepHypRef Expression
1 a9ev 1646 . . 3  |-  E. w  w  =  y
2 ax11inda.1 . . . . . . 7  |-  ( -. 
A. x  x  =  w  ->  ( x  =  w  ->  ( ph  ->  A. x ( x  =  w  ->  ph )
) ) )
32ax11inda2 2151 . . . . . 6  |-  ( -. 
A. x  x  =  w  ->  ( x  =  w  ->  ( A. z ph  ->  A. x
( x  =  w  ->  A. z ph )
) ) )
4 dveeq2-o 2136 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( w  =  y  ->  A. x  w  =  y )
)
54imp 418 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  ->  A. x  w  =  y )
6 hba1-o 2101 . . . . . . . . . 10  |-  ( A. x  w  =  y  ->  A. x A. x  w  =  y )
7 equequ2 1669 . . . . . . . . . . 11  |-  ( w  =  y  ->  (
x  =  w  <->  x  =  y ) )
87sps-o 2111 . . . . . . . . . 10  |-  ( A. x  w  =  y  ->  ( x  =  w  <-> 
x  =  y ) )
96, 8albidh 1580 . . . . . . . . 9  |-  ( A. x  w  =  y  ->  ( A. x  x  =  w  <->  A. x  x  =  y )
)
109notbid 285 . . . . . . . 8  |-  ( A. x  w  =  y  ->  ( -.  A. x  x  =  w  <->  -.  A. x  x  =  y )
)
115, 10syl 15 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( -.  A. x  x  =  w  <->  -.  A. x  x  =  y )
)
127adantl 452 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( x  =  w  <-> 
x  =  y ) )
138imbi1d 308 . . . . . . . . . . 11  |-  ( A. x  w  =  y  ->  ( ( x  =  w  ->  A. z ph )  <->  ( x  =  y  ->  A. z ph ) ) )
146, 13albidh 1580 . . . . . . . . . 10  |-  ( A. x  w  =  y  ->  ( A. x ( x  =  w  ->  A. z ph )  <->  A. x
( x  =  y  ->  A. z ph )
) )
155, 14syl 15 . . . . . . . . 9  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( A. x ( x  =  w  ->  A. z ph )  <->  A. x
( x  =  y  ->  A. z ph )
) )
1615imbi2d 307 . . . . . . . 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 311 . . . . . . 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 311 . . . . . 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 202 . . . . 5  |-  ( ( -.  A. x  x  =  y  /\  w  =  y )  -> 
( -.  A. x  x  =  y  ->  ( x  =  y  -> 
( A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) ) )
2019ex 423 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( w  =  y  ->  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) ) ) )
2120exlimdv 1626 . . 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 16 . 2  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) ) )
2322pm2.43i 43 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-4 2087  ax-5o 2088  ax-6o 2089  ax-10o 2091  ax-12o 2094  ax-16 2096
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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