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Theorem riotaeqbidv 6554
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
Hypotheses
Ref Expression
riotaeqbidv.1  |-  ( ph  ->  A  =  B )
riotaeqbidv.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
riotaeqbidv  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  B ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem riotaeqbidv
StepHypRef Expression
1 riotaeqbidv.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21riotabidv 6553 . 2  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
3 riotaeqbidv.1 . . 3  |-  ( ph  ->  A  =  B )
43riotaeqdv 6552 . 2  |-  ( ph  ->  ( iota_ x  e.  A ch )  =  ( iota_ x  e.  B ch ) )
52, 4eqtrd 2470 1  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  B ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653   iota_crio 6544
This theorem is referenced by:  dfoi  7482  oieq1  7483  oieq2  7484  ordtypecbv  7488  ordtypelem3  7491  zorn2lem1  8378  zorn2g  8385  cidfval  13903  cidval  13904  cidpropd  13938  lubfval  14437  glbfval  14442  spwval2  14658  spwval  14659  grpinvfval  14845  pj1fval  15328  mpfrcl  19941  evlsval  19942  q1pval  20078  ig1pval  20097  gidval  21803  grpoinvfval  21814  pjhfval  22900  cvmliftlem5  24978  cvmliftlem15  24987  trlfset  30959  dicffval  31974  dicfval  31975  dihffval  32030  dihfval  32031  hvmapffval  32558  hvmapfval  32559  hdmap1fval  32597  hdmapffval  32629  hdmapfval  32630  hgmapfval  32689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-riota 6551
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