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Theorem cdleme25cv 31169
Description: Change bound variables in cdleme25c 31166. (Contributed by NM, 2-Feb-2013.)
Hypotheses
Ref Expression
cdleme25cv.f  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme25cv.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )
cdleme25cv.g  |-  G  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
cdleme25cv.o  |-  O  =  ( ( P  .\/  Q )  ./\  ( G  .\/  ( ( R  .\/  z )  ./\  W
) ) )
cdleme25cv.i  |-  I  =  ( iota_ u  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
cdleme25cv.e  |-  E  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
Assertion
Ref Expression
cdleme25cv  |-  I  =  E
Distinct variable groups:    z, s, A    .\/ , s, z    .<_ , s, z    ./\ , s, z    P, s, z    Q, s, z    R, s, z    U, s, z    W, s, z    u, s, z
Allowed substitution hints:    A( u)    B( z, u, s)    P( u)    Q( u)    R( u)    U( u)    E( z, u, s)    F( z, u, s)    G( z, u, s)    I( z, u, s)    .\/ ( u)    .<_ ( u)    ./\ ( u)    N( z, u, s)    O( z, u, s)    W( u)

Proof of Theorem cdleme25cv
StepHypRef Expression
1 breq1 4042 . . . . . . . . 9  |-  ( s  =  z  ->  (
s  .<_  W  <->  z  .<_  W ) )
21notbid 285 . . . . . . . 8  |-  ( s  =  z  ->  ( -.  s  .<_  W  <->  -.  z  .<_  W ) )
3 breq1 4042 . . . . . . . . 9  |-  ( s  =  z  ->  (
s  .<_  ( P  .\/  Q )  <->  z  .<_  ( P 
.\/  Q ) ) )
43notbid 285 . . . . . . . 8  |-  ( s  =  z  ->  ( -.  s  .<_  ( P 
.\/  Q )  <->  -.  z  .<_  ( P  .\/  Q
) ) )
52, 4anbi12d 691 . . . . . . 7  |-  ( s  =  z  ->  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  <->  ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) ) ) )
6 oveq1 5881 . . . . . . . . . . 11  |-  ( s  =  z  ->  (
s  .\/  U )  =  ( z  .\/  U ) )
7 oveq2 5882 . . . . . . . . . . . . 13  |-  ( s  =  z  ->  ( P  .\/  s )  =  ( P  .\/  z
) )
87oveq1d 5889 . . . . . . . . . . . 12  |-  ( s  =  z  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  z )  ./\  W ) )
98oveq2d 5890 . . . . . . . . . . 11  |-  ( s  =  z  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  z ) 
./\  W ) ) )
106, 9oveq12d 5892 . . . . . . . . . 10  |-  ( s  =  z  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) ) )
11 oveq2 5882 . . . . . . . . . . 11  |-  ( s  =  z  ->  ( R  .\/  s )  =  ( R  .\/  z
) )
1211oveq1d 5889 . . . . . . . . . 10  |-  ( s  =  z  ->  (
( R  .\/  s
)  ./\  W )  =  ( ( R 
.\/  z )  ./\  W ) )
1310, 12oveq12d 5892 . . . . . . . . 9  |-  ( s  =  z  ->  (
( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) )  =  ( ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) )
1413oveq2d 5890 . . . . . . . 8  |-  ( s  =  z  ->  (
( P  .\/  Q
)  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) ) )
1514eqeq2d 2307 . . . . . . 7  |-  ( s  =  z  ->  (
u  =  ( ( P  .\/  Q ) 
./\  ( ( ( s  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) )  <->  u  =  ( ( P  .\/  Q )  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) ) ) )
165, 15imbi12d 311 . . . . . 6  |-  ( s  =  z  ->  (
( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q
) )  ->  u  =  ( ( P 
.\/  Q )  ./\  ( ( ( s 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s ) 
./\  W ) ) )  .\/  ( ( R  .\/  s ) 
./\  W ) ) ) )  <->  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) ) ) ) )
1716cbvralv 2777 . . . . 5  |-  ( A. s  e.  A  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  ->  u  =  ( ( P  .\/  Q )  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) ) )  <->  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  ( ( P 
.\/  Q )  ./\  ( ( ( z 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z ) 
./\  W ) ) )  .\/  ( ( R  .\/  z ) 
./\  W ) ) ) ) )
18 cdleme25cv.n . . . . . . . . 9  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )
19 cdleme25cv.f . . . . . . . . . . 11  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
2019oveq1i 5884 . . . . . . . . . 10  |-  ( F 
.\/  ( ( R 
.\/  s )  ./\  W ) )  =  ( ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) )
2120oveq2i 5885 . . . . . . . . 9  |-  ( ( P  .\/  Q ) 
./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) )
2218, 21eqtri 2316 . . . . . . . 8  |-  N  =  ( ( P  .\/  Q )  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) )
2322eqeq2i 2306 . . . . . . 7  |-  ( u  =  N  <->  u  =  ( ( P  .\/  Q )  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) ) )
2423imbi2i 303 . . . . . 6  |-  ( ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  ->  u  =  N )  <->  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) ) ) )
2524ralbii 2580 . . . . 5  |-  ( A. s  e.  A  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  ->  u  =  N )  <->  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) ) ) )
26 cdleme25cv.o . . . . . . . . 9  |-  O  =  ( ( P  .\/  Q )  ./\  ( G  .\/  ( ( R  .\/  z )  ./\  W
) ) )
27 cdleme25cv.g . . . . . . . . . . 11  |-  G  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
2827oveq1i 5884 . . . . . . . . . 10  |-  ( G 
.\/  ( ( R 
.\/  z )  ./\  W ) )  =  ( ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) )
2928oveq2i 5885 . . . . . . . . 9  |-  ( ( P  .\/  Q ) 
./\  ( G  .\/  ( ( R  .\/  z )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) )
3026, 29eqtri 2316 . . . . . . . 8  |-  O  =  ( ( P  .\/  Q )  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) )
3130eqeq2i 2306 . . . . . . 7  |-  ( u  =  O  <->  u  =  ( ( P  .\/  Q )  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) ) )
3231imbi2i 303 . . . . . 6  |-  ( ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q ) )  ->  u  =  O )  <->  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) ) ) )
3332ralbii 2580 . . . . 5  |-  ( A. z  e.  A  (
( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q ) )  ->  u  =  O )  <->  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) ) ) )
3417, 25, 333bitr4i 268 . . . 4  |-  ( A. s  e.  A  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  ->  u  =  N )  <->  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  O ) )
3534a1i 10 . . 3  |-  ( u  e.  B  ->  ( A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q
) )  ->  u  =  N )  <->  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  O ) ) )
3635riotabiia 6338 . 2  |-  ( iota_ u  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  N ) )  =  (
iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
37 cdleme25cv.i . 2  |-  I  =  ( iota_ u  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
38 cdleme25cv.e . 2  |-  E  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
3936, 37, 383eqtr4i 2326 1  |-  I  =  E
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039  (class class class)co 5874   iota_crio 6313
This theorem is referenced by:  cdleme27a  31178
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-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-riota 6320
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