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Theorem riotaeqbidv 6249
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 6248 . 2  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  A  ch ) )
3 riotaeqbidv.1 . . 3  |-  ( ph  ->  A  =  B )
43riotaeqdv 6247 . 2  |-  ( ph  ->  ( iota_ x  e.  A  ch )  =  ( iota_ x  e.  B  ch ) )
52, 4eqtrd 2508 1  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  B  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   iota_crio 6245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-uni 4246  df-iota 5551  df-riota 6246
This theorem is referenced by:  dfoi  7937  oieq1  7938  oieq2  7939  ordtypecbv  7943  ordtypelem3  7946  zorn2lem1  8877  zorn2g  8884  cidfval  14934  cidval  14935  cidpropd  14969  lubfval  15468  glbfval  15481  grpinvfval  15902  pj1fval  16527  mpfrcl  17998  evlsval  17999  q1pval  22381  ig1pval  22400  mirval  23846  midf  23916  ismidb  23918  lmif  23925  islmib  23927  gidval  24988  grpoinvfval  24999  pjhfval  26087  cvmliftlem5  28485  cvmliftlem15  28494  trlfset  35173  dicffval  36188  dicfval  36189  dihffval  36244  dihfval  36245  hvmapffval  36772  hvmapfval  36773  hdmap1fval  36811  hdmapffval  36843  hdmapfval  36844  hgmapfval  36903
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