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Theorem riotasv3d 6353
Description: A property  ch holding for a representative of a single-valued class expression  C ( y ) (see e.g. reusv2 4540) 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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 riotasv3d.7 . . . 4  |-  ( ph  ->  E. y  e.  B  ps )
32adantr 451 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  E. y  e.  B  ps )
4 riotasv3d.1 . . . . . 6  |-  F/ y
ph
5 nfv 1605 . . . . . 6  |-  F/ y  A  e.  _V
6 riotasv3d.5 . . . . . . . . . 10  |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )
76imp 418 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  ps )
)  ->  ch )
87adantrl 696 . . . . . . . 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 6347 . . . . . . . . . . 11  |-  ( (
ph  /\  A  e.  _V )  ->  ( ( y  e.  B  /\  ps )  ->  D  =  C ) )
1211impr 602 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  D  =  C )
1312eqcomd 2288 . . . . . . . . 9  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  C  =  D )
14 riotasv3d.4 . . . . . . . . 9  |-  ( (
ph  /\  C  =  D )  ->  ( ch 
<->  th ) )
1513, 14syldan 456 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  -> 
( ch  <->  th )
)
168, 15mpbid 201 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  th )
1716exp45 597 . . . . . 6  |-  ( ph  ->  ( A  e.  _V  ->  ( y  e.  B  ->  ( ps  ->  th )
) ) )
184, 5, 17ralrimd 2631 . . . . 5  |-  ( ph  ->  ( A  e.  _V  ->  A. y  e.  B  ( ps  ->  th )
) )
19 riotasv3d.2 . . . . . 6  |-  ( ph  ->  F/ y th )
20 r19.23t 2657 . . . . . 6  |-  ( F/ y th  ->  ( A. y  e.  B  ( ps  ->  th )  <->  ( E. y  e.  B  ps  ->  th ) ) )
2119, 20syl 15 . . . . 5  |-  ( ph  ->  ( A. y  e.  B  ( ps  ->  th )  <->  ( E. y  e.  B  ps  ->  th ) ) )
2218, 21sylibd 205 . . . 4  |-  ( ph  ->  ( A  e.  _V  ->  ( E. y  e.  B  ps  ->  th )
) )
2322imp 418 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  ( E. y  e.  B  ps  ->  th ) )
243, 23mpd 14 . 2  |-  ( (
ph  /\  A  e.  _V )  ->  th )
251, 24sylan2 460 1  |-  ( (
ph  /\  A  e.  V )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1531    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   iota_crio 6297
This theorem is referenced by:  cdlemefs32sn1aw  30603  cdleme43fsv1snlem  30609  cdleme41sn3a  30622  cdleme40m  30656  cdleme40n  30657  cdlemkid  31125  dihvalcqpre  31425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-undef 6298  df-riota 6304
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