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Theorem unirep 28632
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 2444 . . . . 5  |-  ( ps 
->  C  =  C
)
21ancli 551 . . . 4  |-  ( ps 
->  ( ps  /\  C  =  C ) )
3 unirep.1 . . . . . 6  |-  ( y  =  D  ->  ( ph 
<->  ps ) )
4 unirep.2 . . . . . . 7  |-  ( y  =  D  ->  B  =  C )
54eqeq2d 2454 . . . . . 6  |-  ( y  =  D  ->  ( C  =  B  <->  C  =  C ) )
63, 5anbi12d 710 . . . . 5  |-  ( y  =  D  ->  (
( ph  /\  C  =  B )  <->  ( ps  /\  C  =  C ) ) )
76rspcev 3094 . . . 4  |-  ( ( D  e.  A  /\  ( ps  /\  C  =  C ) )  ->  E. y  e.  A  ( ph  /\  C  =  B ) )
82, 7sylan2 474 . . 3  |-  ( ( D  e.  A  /\  ps )  ->  E. y  e.  A  ( ph  /\  C  =  B ) )
98adantl 466 . 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 2590 . . . . . 6  |-  ( D  e.  A  ->  F/_ y C )
1110, 4csbiegf 3333 . . . . 5  |-  ( D  e.  A  ->  [_ D  /  y ]_ B  =  C )
12 unirep.5 . . . . . 6  |-  B  e. 
_V
1312csbex 4446 . . . . 5  |-  [_ D  /  y ]_ B  e.  _V
1411, 13syl6eqelr 2532 . . . 4  |-  ( D  e.  A  ->  C  e.  _V )
1514ad2antrl 727 . . 3  |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  C  e.  _V )
16 eqeq1 2449 . . . . . . . . . . 11  |-  ( x  =  C  ->  (
x  =  B  <->  C  =  B ) )
1716anbi2d 703 . . . . . . . . . 10  |-  ( x  =  C  ->  (
( ph  /\  x  =  B )  <->  ( ph  /\  C  =  B ) ) )
1817rexbidv 2757 . . . . . . . . 9  |-  ( x  =  C  ->  ( E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ( ph  /\  C  =  B ) ) )
1918spcegv 3079 . . . . . . . 8  |-  ( C  e.  _V  ->  ( E. y  e.  A  ( ph  /\  C  =  B )  ->  E. x E. y  e.  A  ( ph  /\  x  =  B ) ) )
2014, 19syl 16 . . . . . . 7  |-  ( D  e.  A  ->  ( E. y  e.  A  ( ph  /\  C  =  B )  ->  E. x E. y  e.  A  ( ph  /\  x  =  B ) ) )
2120adantr 465 . . . . . 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 466 . . . 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 2878 . . . . . . . 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 2878 . . . . . . . . . . . 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 820 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  =  B )  /\  ( ch  /\  w  =  F ) )  <->  ( ( ph  /\  ch )  /\  ( x  =  B  /\  w  =  F
) ) )
27 pm3.35 587 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  ch )  /\  ( ( ph  /\ 
ch )  ->  B  =  F ) )  ->  B  =  F )
28 eqeq12 2455 . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ch )  ->  B  =  F )  /\  ( ph  /\  ch ) )  ->  ( ( x  =  B  /\  w  =  F )  ->  x  =  w ) )
3130expimpd 603 . . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ch )  ->  B  =  F )  ->  ( (
( ch  /\  w  =  F )  /\  ( ph  /\  x  =  B ) )  ->  x  =  w ) )
3433expdimp 437 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ch )  ->  B  =  F )  /\  ( ch  /\  w  =  F ) )  ->  (
( ph  /\  x  =  B )  ->  x  =  w ) )
3534rexlimivw 2858 . . . . . . . . . . . . 13  |-  ( E. z  e.  A  ( ( ( ph  /\  ch )  ->  B  =  F )  /\  ( ch  /\  w  =  F ) )  ->  (
( ph  /\  x  =  B )  ->  x  =  w ) )
3635imp 429 . . . . . . . . . . . 12  |-  ( ( E. z  e.  A  ( ( ( ph  /\ 
ch )  ->  B  =  F )  /\  ( ch  /\  w  =  F ) )  /\  ( ph  /\  x  =  B ) )  ->  x  =  w )
3725, 36sylan 471 . . . . . . . . . . 11  |-  ( ( ( A. z  e.  A  ( ( ph  /\ 
ch )  ->  B  =  F )  /\  E. z  e.  A  ( ch  /\  w  =  F ) )  /\  ( ph  /\  x  =  B ) )  ->  x  =  w )
3837an32s 802 . . . . . . . . . 10  |-  ( ( ( A. z  e.  A  ( ( ph  /\ 
ch )  ->  B  =  F )  /\  ( ph  /\  x  =  B ) )  /\  E. z  e.  A  ( ch  /\  w  =  F ) )  ->  x  =  w )
3938ex 434 . . . . . . . . 9  |-  ( ( A. z  e.  A  ( ( ph  /\  ch )  ->  B  =  F )  /\  ( ph  /\  x  =  B ) )  ->  ( E. z  e.  A  ( ch  /\  w  =  F )  ->  x  =  w ) )
4039rexlimivw 2858 . . . . . . . 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 16 . . . . . . 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 603 . . . . . 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 465 . . . . 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 1686 . . . 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 2449 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  B  <->  w  =  B ) )
4645anbi2d 703 . . . . . . 7  |-  ( x  =  w  ->  (
( ph  /\  x  =  B )  <->  ( ph  /\  w  =  B ) ) )
4746rexbidv 2757 . . . . . 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 2454 . . . . . . . 8  |-  ( y  =  z  ->  (
w  =  B  <->  w  =  F ) )
5148, 50anbi12d 710 . . . . . . 7  |-  ( y  =  z  ->  (
( ph  /\  w  =  B )  <->  ( ch  /\  w  =  F ) ) )
5251cbvrexv 2969 . . . . . 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 2318 . . . 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 664 . . 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 5428 . . 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 661 . 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 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253   A.wral 2736   E.wrex 2737   _Vcvv 2993   [_csb 3309   iotacio 5400
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-ext 2423  ax-nul 4442
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-sn 3899  df-pr 3901  df-uni 4113  df-iota 5402
This theorem is referenced by: (None)
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