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Theorem riotaxfrd 6195
Description: Change the variable  x in the expression for "the unique  x such that  ps " to another variable  y contained in expression  B. Use reuhypd 4630 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riotaxfrd.1  |-  F/_ y C
riotaxfrd.2  |-  ( (
ph  /\  y  e.  A )  ->  B  e.  A )
riotaxfrd.3  |-  ( (
ph  /\  ( iota_ y  e.  A  ch )  e.  A )  ->  C  e.  A )
riotaxfrd.4  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
riotaxfrd.5  |-  ( y  =  ( iota_ y  e.  A  ch )  ->  B  =  C )
riotaxfrd.6  |-  ( (
ph  /\  x  e.  A )  ->  E! y  e.  A  x  =  B )
Assertion
Ref Expression
riotaxfrd  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A  ps )  =  C
)
Distinct variable groups:    x, B    x, C    x, y, A    ph, x, y    ps, y    ch, x
Allowed substitution hints:    ps( x)    ch( y)    B( y)    C( y)

Proof of Theorem riotaxfrd
StepHypRef Expression
1 rabid 3003 . . . 4  |-  ( x  e.  { x  e.  A  |  ps }  <->  ( x  e.  A  /\  ps ) )
21baib 896 . . 3  |-  ( x  e.  A  ->  (
x  e.  { x  e.  A  |  ps } 
<->  ps ) )
32riotabiia 6182 . 2  |-  ( iota_ x  e.  A  x  e. 
{ x  e.  A  |  ps } )  =  ( iota_ x  e.  A  ps )
4 riotaxfrd.2 . . . . . 6  |-  ( (
ph  /\  y  e.  A )  ->  B  e.  A )
5 riotaxfrd.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  E! y  e.  A  x  =  B )
6 riotaxfrd.4 . . . . . 6  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
74, 5, 6reuxfrd 4628 . . . . 5  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! y  e.  A  ch )
)
8 riotacl2 6178 . . . . . . . 8  |-  ( E! y  e.  A  ch  ->  ( iota_ y  e.  A  ch )  e.  { y  e.  A  |  ch } )
98adantl 466 . . . . . . 7  |-  ( (
ph  /\  E! y  e.  A  ch )  ->  ( iota_ y  e.  A  ch )  e.  { y  e.  A  |  ch } )
10 riotacl 6179 . . . . . . . 8  |-  ( E! y  e.  A  ch  ->  ( iota_ y  e.  A  ch )  e.  A
)
11 nfriota1 6171 . . . . . . . . 9  |-  F/_ y
( iota_ y  e.  A  ch )
12 riotaxfrd.1 . . . . . . . . 9  |-  F/_ y C
13 riotaxfrd.5 . . . . . . . . 9  |-  ( y  =  ( iota_ y  e.  A  ch )  ->  B  =  C )
1411, 12, 4, 6, 13rabxfrd 4624 . . . . . . . 8  |-  ( (
ph  /\  ( iota_ y  e.  A  ch )  e.  A )  ->  ( C  e.  { x  e.  A  |  ps } 
<->  ( iota_ y  e.  A  ch )  e.  { y  e.  A  |  ch } ) )
1510, 14sylan2 474 . . . . . . 7  |-  ( (
ph  /\  E! y  e.  A  ch )  ->  ( C  e.  {
x  e.  A  |  ps }  <->  ( iota_ y  e.  A  ch )  e. 
{ y  e.  A  |  ch } ) )
169, 15mpbird 232 . . . . . 6  |-  ( (
ph  /\  E! y  e.  A  ch )  ->  C  e.  { x  e.  A  |  ps } )
1716ex 434 . . . . 5  |-  ( ph  ->  ( E! y  e.  A  ch  ->  C  e.  { x  e.  A  |  ps } ) )
187, 17sylbid 215 . . . 4  |-  ( ph  ->  ( E! x  e.  A  ps  ->  C  e.  { x  e.  A  |  ps } ) )
1918imp 429 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  C  e.  { x  e.  A  |  ps } )
20 riotaxfrd.3 . . . . . . . 8  |-  ( (
ph  /\  ( iota_ y  e.  A  ch )  e.  A )  ->  C  e.  A )
2120ex 434 . . . . . . 7  |-  ( ph  ->  ( ( iota_ y  e.  A  ch )  e.  A  ->  C  e.  A ) )
2210, 21syl5 32 . . . . . 6  |-  ( ph  ->  ( E! y  e.  A  ch  ->  C  e.  A ) )
237, 22sylbid 215 . . . . 5  |-  ( ph  ->  ( E! x  e.  A  ps  ->  C  e.  A ) )
2423imp 429 . . . 4  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  C  e.  A )
251baibr 897 . . . . . . 7  |-  ( x  e.  A  ->  ( ps 
<->  x  e.  { x  e.  A  |  ps } ) )
2625reubiia 3012 . . . . . 6  |-  ( E! x  e.  A  ps  <->  E! x  e.  A  x  e.  { x  e.  A  |  ps }
)
2726biimpi 194 . . . . 5  |-  ( E! x  e.  A  ps  ->  E! x  e.  A  x  e.  { x  e.  A  |  ps } )
2827adantl 466 . . . 4  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x  e.  A  x  e.  { x  e.  A  |  ps } )
29 nfcv 2616 . . . . 5  |-  F/_ x C
30 nfrab1 3007 . . . . . 6  |-  F/_ x { x  e.  A  |  ps }
3130nfel2 2634 . . . . 5  |-  F/ x  C  e.  { x  e.  A  |  ps }
32 eleq1 2526 . . . . 5  |-  ( x  =  C  ->  (
x  e.  { x  e.  A  |  ps } 
<->  C  e.  { x  e.  A  |  ps } ) )
3329, 31, 32riota2f 6186 . . . 4  |-  ( ( C  e.  A  /\  E! x  e.  A  x  e.  { x  e.  A  |  ps } )  ->  ( C  e.  { x  e.  A  |  ps } 
<->  ( iota_ x  e.  A  x  e.  { x  e.  A  |  ps } )  =  C ) )
3424, 28, 33syl2anc 661 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( C  e.  {
x  e.  A  |  ps }  <->  ( iota_ x  e.  A  x  e.  {
x  e.  A  |  ps } )  =  C ) )
3519, 34mpbid 210 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A  x  e.  { x  e.  A  |  ps } )  =  C )
363, 35syl5eqr 2509 1  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A  ps )  =  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   F/_wnfc 2602   E!wreu 2801   {crab 2803   iota_crio 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-un 3444  df-in 3446  df-ss 3453  df-sn 3989  df-pr 3991  df-uni 4203  df-iota 5492  df-riota 6164
This theorem is referenced by:  riotaneg  10419  zriotaneg  10868  riotaocN  33212
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