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Theorem unirep 31449
Description: Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice. (Contributed by Jeff Madsen, 1-Jun-2011.)
Hypotheses
Ref Expression
unirep.1  |-  ( y  =  D  ->  ( ph 
<->  ps ) )
unirep.2  |-  ( y  =  D  ->  B  =  C )
unirep.3  |-  ( y  =  z  ->  ( ph 
<->  ch ) )
unirep.4  |-  ( y  =  z  ->  B  =  F )
unirep.5  |-  B  e. 
_V
Assertion
Ref Expression
unirep  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  ( iota x E. y  e.  A  ( ph  /\  x  =  B )
)  =  C )
Distinct variable groups:    x, A, y, z    x, B, z   
x, C, y    x, D, y    x, F, y    ph, x, z    ps, x, y    ch, x, y
Allowed substitution hints:    ph( y)    ps( z)    ch( z)    B( y)    C( z)    D( z)    F( z)

Proof of Theorem unirep
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 eqidd 2401 . . . . 5  |-  ( ps 
->  C  =  C
)
21ancli 549 . . . 4  |-  ( ps 
->  ( ps  /\  C  =  C ) )
3 unirep.1 . . . . . 6  |-  ( y  =  D  ->  ( ph 
<->  ps ) )
4 unirep.2 . . . . . . 7  |-  ( y  =  D  ->  B  =  C )
54eqeq2d 2414 . . . . . 6  |-  ( y  =  D  ->  ( C  =  B  <->  C  =  C ) )
63, 5anbi12d 709 . . . . 5  |-  ( y  =  D  ->  (
( ph  /\  C  =  B )  <->  ( ps  /\  C  =  C ) ) )
76rspcev 3157 . . . 4  |-  ( ( D  e.  A  /\  ( ps  /\  C  =  C ) )  ->  E. y  e.  A  ( ph  /\  C  =  B ) )
82, 7sylan2 472 . . 3  |-  ( ( D  e.  A  /\  ps )  ->  E. y  e.  A  ( ph  /\  C  =  B ) )
98adantl 464 . 2  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  E. y  e.  A  ( ph  /\  C  =  B ) )
10 nfcvd 2563 . . . . . 6  |-  ( D  e.  A  ->  F/_ y C )
1110, 4csbiegf 3394 . . . . 5  |-  ( D  e.  A  ->  [_ D  /  y ]_ B  =  C )
12 unirep.5 . . . . . 6  |-  B  e. 
_V
1312csbex 4526 . . . . 5  |-  [_ D  /  y ]_ B  e.  _V
1411, 13syl6eqelr 2497 . . . 4  |-  ( D  e.  A  ->  C  e.  _V )
1514ad2antrl 726 . . 3  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  C  e.  _V )
16 eqeq1 2404 . . . . . . . . . . 11  |-  ( x  =  C  ->  (
x  =  B  <->  C  =  B ) )
1716anbi2d 702 . . . . . . . . . 10  |-  ( x  =  C  ->  (
( ph  /\  x  =  B )  <->  ( ph  /\  C  =  B ) ) )
1817rexbidv 2915 . . . . . . . . 9  |-  ( x  =  C  ->  ( E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ( ph  /\  C  =  B ) ) )
1918spcegv 3142 . . . . . . . 8  |-  ( C  e.  _V  ->  ( E. y  e.  A  ( ph  /\  C  =  B )  ->  E. x E. y  e.  A  ( ph  /\  x  =  B ) ) )
2014, 19syl 17 . . . . . . 7  |-  ( D  e.  A  ->  ( E. y  e.  A  ( ph  /\  C  =  B )  ->  E. x E. y  e.  A  ( ph  /\  x  =  B ) ) )
2120adantr 463 . . . . . 6  |-  ( ( D  e.  A  /\  ps )  ->  ( E. y  e.  A  (
ph  /\  C  =  B )  ->  E. x E. y  e.  A  ( ph  /\  x  =  B ) ) )
228, 21mpd 15 . . . . 5  |-  ( ( D  e.  A  /\  ps )  ->  E. x E. y  e.  A  ( ph  /\  x  =  B ) )
2322adantl 464 . . . 4  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  E. x E. y  e.  A  ( ph  /\  x  =  B ) )
24 r19.29 2939 . . . . . . . 8  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  E. y  e.  A  ( ph  /\  x  =  B ) )  ->  E. y  e.  A  ( A. z  e.  A  (
( ph  /\  ch )  ->  B  =  F )  /\  ( ph  /\  x  =  B )
) )
25 r19.29 2939 . . . . . . . . . . . 12  |-  ( ( A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  E. z  e.  A  ( ch  /\  w  =  F ) )  ->  E. z  e.  A  ( (
( ph  /\  ch )  ->  B  =  F )  /\  ( ch  /\  w  =  F )
) )
26 an4 823 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  =  B )  /\  ( ch  /\  w  =  F ) )  <->  ( ( ph  /\  ch )  /\  ( x  =  B  /\  w  =  F
) ) )
27 pm3.35 585 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  ch )  /\  ( ( ph  /\ 
ch )  ->  B  =  F ) )  ->  B  =  F )
28 eqeq12 2419 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  B  /\  w  =  F )  ->  ( x  =  w  <-> 
B  =  F ) )
2927, 28syl5ibrcom 222 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  ch )  /\  ( ( ph  /\ 
ch )  ->  B  =  F ) )  -> 
( ( x  =  B  /\  w  =  F )  ->  x  =  w ) )
3029ancoms 451 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ch )  ->  B  =  F )  /\  ( ph  /\  ch ) )  ->  ( ( x  =  B  /\  w  =  F )  ->  x  =  w ) )
3130expimpd 601 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ch )  ->  B  =  F )  ->  ( (
( ph  /\  ch )  /\  ( x  =  B  /\  w  =  F ) )  ->  x  =  w ) )
3226, 31syl5bi 217 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ch )  ->  B  =  F )  ->  ( (
( ph  /\  x  =  B )  /\  ( ch  /\  w  =  F ) )  ->  x  =  w ) )
3332ancomsd 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ch )  ->  B  =  F )  ->  ( (
( ch  /\  w  =  F )  /\  ( ph  /\  x  =  B ) )  ->  x  =  w ) )
3433expdimp 435 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ch )  ->  B  =  F )  /\  ( ch  /\  w  =  F ) )  ->  (
( ph  /\  x  =  B )  ->  x  =  w ) )
3534rexlimivw 2890 . . . . . . . . . . . . 13  |-  ( E. z  e.  A  ( ( ( ph  /\  ch )  ->  B  =  F )  /\  ( ch  /\  w  =  F ) )  ->  (
( ph  /\  x  =  B )  ->  x  =  w ) )
3635imp 427 . . . . . . . . . . . 12  |-  ( ( E. z  e.  A  ( ( ( ph  /\ 
ch )  ->  B  =  F )  /\  ( ch  /\  w  =  F ) )  /\  ( ph  /\  x  =  B ) )  ->  x  =  w )
3725, 36sylan 469 . . . . . . . . . . 11  |-  ( ( ( A. z  e.  A  ( ( ph  /\ 
ch )  ->  B  =  F )  /\  E. z  e.  A  ( ch  /\  w  =  F ) )  /\  ( ph  /\  x  =  B ) )  ->  x  =  w )
3837an32s 803 . . . . . . . . . 10  |-  ( ( ( A. z  e.  A  ( ( ph  /\ 
ch )  ->  B  =  F )  /\  ( ph  /\  x  =  B ) )  /\  E. z  e.  A  ( ch  /\  w  =  F ) )  ->  x  =  w )
3938ex 432 . . . . . . . . 9  |-  ( ( A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( ph  /\  x  =  B ) )  ->  ( E. z  e.  A  ( ch  /\  w  =  F )  ->  x  =  w ) )
4039rexlimivw 2890 . . . . . . . 8  |-  ( E. y  e.  A  ( A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( ph  /\  x  =  B ) )  ->  ( E. z  e.  A  ( ch  /\  w  =  F )  ->  x  =  w ) )
4124, 40syl 17 . . . . . . 7  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  E. y  e.  A  ( ph  /\  x  =  B ) )  ->  ( E. z  e.  A  ( ch  /\  w  =  F )  ->  x  =  w ) )
4241expimpd 601 . . . . . 6  |-  ( A. y  e.  A  A. z  e.  A  (
( ph  /\  ch )  ->  B  =  F )  ->  ( ( E. y  e.  A  (
ph  /\  x  =  B )  /\  E. z  e.  A  ( ch  /\  w  =  F ) )  ->  x  =  w ) )
4342adantr 463 . . . . 5  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  (
( E. y  e.  A  ( ph  /\  x  =  B )  /\  E. z  e.  A  ( ch  /\  w  =  F ) )  ->  x  =  w )
)
4443alrimivv 1739 . . . 4  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  A. x A. w ( ( E. y  e.  A  (
ph  /\  x  =  B )  /\  E. z  e.  A  ( ch  /\  w  =  F ) )  ->  x  =  w ) )
45 eqeq1 2404 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  B  <->  w  =  B ) )
4645anbi2d 702 . . . . . . 7  |-  ( x  =  w  ->  (
( ph  /\  x  =  B )  <->  ( ph  /\  w  =  B ) ) )
4746rexbidv 2915 . . . . . 6  |-  ( x  =  w  ->  ( E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ( ph  /\  w  =  B ) ) )
48 unirep.3 . . . . . . . 8  |-  ( y  =  z  ->  ( ph 
<->  ch ) )
49 unirep.4 . . . . . . . . 9  |-  ( y  =  z  ->  B  =  F )
5049eqeq2d 2414 . . . . . . . 8  |-  ( y  =  z  ->  (
w  =  B  <->  w  =  F ) )
5148, 50anbi12d 709 . . . . . . 7  |-  ( y  =  z  ->  (
( ph  /\  w  =  B )  <->  ( ch  /\  w  =  F ) ) )
5251cbvrexv 3032 . . . . . 6  |-  ( E. y  e.  A  (
ph  /\  w  =  B )  <->  E. z  e.  A  ( ch  /\  w  =  F ) )
5347, 52syl6bb 261 . . . . 5  |-  ( x  =  w  ->  ( E. y  e.  A  ( ph  /\  x  =  B )  <->  E. z  e.  A  ( ch  /\  w  =  F ) ) )
5453eu4 2288 . . . 4  |-  ( E! x E. y  e.  A  ( ph  /\  x  =  B )  <->  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  /\  A. x A. w ( ( E. y  e.  A  (
ph  /\  x  =  B )  /\  E. z  e.  A  ( ch  /\  w  =  F ) )  ->  x  =  w ) ) )
5523, 44, 54sylanbrc 662 . . 3  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  E! x E. y  e.  A  ( ph  /\  x  =  B ) )
5618iota2 5513 . . 3  |-  ( ( C  e.  _V  /\  E! x E. y  e.  A  ( ph  /\  x  =  B )
)  ->  ( E. y  e.  A  ( ph  /\  C  =  B )  <->  ( iota x E. y  e.  A  ( ph  /\  x  =  B ) )  =  C ) )
5715, 55, 56syl2anc 659 . 2  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  ( E. y  e.  A  ( ph  /\  C  =  B )  <->  ( iota x E. y  e.  A  ( ph  /\  x  =  B ) )  =  C ) )
589, 57mpbid 210 1  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  ( iota x E. y  e.  A  ( ph  /\  x  =  B )
)  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1401    = wceq 1403   E.wex 1631    e. wcel 1840   E!weu 2236   A.wral 2751   E.wrex 2752   _Vcvv 3056   [_csb 3370   iotacio 5485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-nul 4522
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-sn 3970  df-pr 3972  df-uni 4189  df-iota 5487
This theorem is referenced by: (None)
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