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Theorem cdleme25cv 36206
Description: Change bound variables in cdleme25c 36203. (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 4459 . . . . . . . . 9  |-  ( s  =  z  ->  (
s  .<_  W  <->  z  .<_  W ) )
21notbid 294 . . . . . . . 8  |-  ( s  =  z  ->  ( -.  s  .<_  W  <->  -.  z  .<_  W ) )
3 breq1 4459 . . . . . . . . 9  |-  ( s  =  z  ->  (
s  .<_  ( P  .\/  Q )  <->  z  .<_  ( P 
.\/  Q ) ) )
43notbid 294 . . . . . . . 8  |-  ( s  =  z  ->  ( -.  s  .<_  ( P 
.\/  Q )  <->  -.  z  .<_  ( P  .\/  Q
) ) )
52, 4anbi12d 710 . . . . . . 7  |-  ( s  =  z  ->  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  <->  ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) ) ) )
6 oveq1 6303 . . . . . . . . . . 11  |-  ( s  =  z  ->  (
s  .\/  U )  =  ( z  .\/  U ) )
7 oveq2 6304 . . . . . . . . . . . . 13  |-  ( s  =  z  ->  ( P  .\/  s )  =  ( P  .\/  z
) )
87oveq1d 6311 . . . . . . . . . . . 12  |-  ( s  =  z  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  z )  ./\  W ) )
98oveq2d 6312 . . . . . . . . . . 11  |-  ( s  =  z  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  z ) 
./\  W ) ) )
106, 9oveq12d 6314 . . . . . . . . . 10  |-  ( s  =  z  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) ) )
11 oveq2 6304 . . . . . . . . . . 11  |-  ( s  =  z  ->  ( R  .\/  s )  =  ( R  .\/  z
) )
1211oveq1d 6311 . . . . . . . . . 10  |-  ( s  =  z  ->  (
( R  .\/  s
)  ./\  W )  =  ( ( R 
.\/  z )  ./\  W ) )
1310, 12oveq12d 6314 . . . . . . . . 9  |-  ( s  =  z  ->  (
( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) )  =  ( ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) )
1413oveq2d 6312 . . . . . . . 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 2471 . . . . . . 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 320 . . . . . 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 3084 . . . . 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 6306 . . . . . . . . . 10  |-  ( F 
.\/  ( ( R 
.\/  s )  ./\  W ) )  =  ( ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) )
2120oveq2i 6307 . . . . . . . . 9  |-  ( ( P  .\/  Q ) 
./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) )
2218, 21eqtri 2486 . . . . . . . 8  |-  N  =  ( ( P  .\/  Q )  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) )
2322eqeq2i 2475 . . . . . . 7  |-  ( u  =  N  <->  u  =  ( ( P  .\/  Q )  ./\  ( (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  .\/  ( ( R  .\/  s )  ./\  W
) ) ) )
2423imbi2i 312 . . . . . 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 2888 . . . . 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 6306 . . . . . . . . . 10  |-  ( G 
.\/  ( ( R 
.\/  z )  ./\  W ) )  =  ( ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) )
2928oveq2i 6307 . . . . . . . . 9  |-  ( ( P  .\/  Q ) 
./\  ( G  .\/  ( ( R  .\/  z )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) )
3026, 29eqtri 2486 . . . . . . . 8  |-  O  =  ( ( P  .\/  Q )  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) )
3130eqeq2i 2475 . . . . . . 7  |-  ( u  =  O  <->  u  =  ( ( P  .\/  Q )  ./\  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  ( ( R  .\/  z )  ./\  W
) ) ) )
3231imbi2i 312 . . . . . 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 2888 . . . . 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 277 . . . 4  |-  ( A. s  e.  A  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  ->  u  =  N )  <->  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  O ) )
3534a1i 11 . . 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 6275 . 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 2496 1  |-  I  =  E
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   class class class wbr 4456   iota_crio 6257  (class class class)co 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-riota 6258  df-ov 6299
This theorem is referenced by:  cdleme27a  36215
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