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Theorem riotasv3d 6598
Description: A property  ch holding for a representative of a single-valued class expression  C ( y ) (see e.g. reusv2 4729) also holds for its description binder  D (in the form of property  th). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riotasv3d.1  |-  F/ y
ph
riotasv3d.2  |-  ( ph  ->  F/ y th )
riotasv3d.3  |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )
riotasv3d.4  |-  ( (
ph  /\  C  =  D )  ->  ( ch 
<->  th ) )
riotasv3d.5  |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )
riotasv3d.6  |-  ( ph  ->  D  e.  A )
riotasv3d.7  |-  ( ph  ->  E. y  e.  B  ps )
Assertion
Ref Expression
riotasv3d  |-  ( (
ph  /\  A  e.  V )  ->  th )
Distinct variable groups:    x, y, A    x, B    x, C    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x, y)    th( x, y)    B( y)    C( y)    D( x, y)    V( x, y)

Proof of Theorem riotasv3d
StepHypRef Expression
1 elex 2964 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 riotasv3d.7 . . . 4  |-  ( ph  ->  E. y  e.  B  ps )
32adantr 452 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  E. y  e.  B  ps )
4 riotasv3d.1 . . . . . 6  |-  F/ y
ph
5 nfv 1629 . . . . . 6  |-  F/ y  A  e.  _V
6 riotasv3d.5 . . . . . . . . . 10  |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )
76imp 419 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  ps )
)  ->  ch )
87adantrl 697 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  ch )
9 riotasv3d.3 . . . . . . . . . . . 12  |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )
10 riotasv3d.6 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  A )
119, 10riotasvd 6592 . . . . . . . . . . 11  |-  ( (
ph  /\  A  e.  _V )  ->  ( ( y  e.  B  /\  ps )  ->  D  =  C ) )
1211impr 603 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  D  =  C )
1312eqcomd 2441 . . . . . . . . 9  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  C  =  D )
14 riotasv3d.4 . . . . . . . . 9  |-  ( (
ph  /\  C  =  D )  ->  ( ch 
<->  th ) )
1513, 14syldan 457 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  -> 
( ch  <->  th )
)
168, 15mpbid 202 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  th )
1716exp45 598 . . . . . 6  |-  ( ph  ->  ( A  e.  _V  ->  ( y  e.  B  ->  ( ps  ->  th )
) ) )
184, 5, 17ralrimd 2794 . . . . 5  |-  ( ph  ->  ( A  e.  _V  ->  A. y  e.  B  ( ps  ->  th )
) )
19 riotasv3d.2 . . . . . 6  |-  ( ph  ->  F/ y th )
20 r19.23t 2820 . . . . . 6  |-  ( F/ y th  ->  ( A. y  e.  B  ( ps  ->  th )  <->  ( E. y  e.  B  ps  ->  th ) ) )
2119, 20syl 16 . . . . 5  |-  ( ph  ->  ( A. y  e.  B  ( ps  ->  th )  <->  ( E. y  e.  B  ps  ->  th ) ) )
2218, 21sylibd 206 . . . 4  |-  ( ph  ->  ( A  e.  _V  ->  ( E. y  e.  B  ps  ->  th )
) )
2322imp 419 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  ( E. y  e.  B  ps  ->  th ) )
243, 23mpd 15 . 2  |-  ( (
ph  /\  A  e.  _V )  ->  th )
251, 24sylan2 461 1  |-  ( (
ph  /\  A  e.  V )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   _Vcvv 2956   iota_crio 6542
This theorem is referenced by:  cdlemefs32sn1aw  31211  cdleme43fsv1snlem  31217  cdleme41sn3a  31230  cdleme40m  31264  cdleme40n  31265  cdlemkid  31733  dihvalcqpre  32033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-undef 6543  df-riota 6549
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