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Theorem riotaeqbidv 6323
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 6322 . 2  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
3 riotaeqbidv.1 . . 3  |-  ( ph  ->  A  =  B )
43riotaeqdv 6321 . 2  |-  ( ph  ->  ( iota_ x  e.  A ch )  =  ( iota_ x  e.  B ch ) )
52, 4eqtrd 2328 1  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  B ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632   iota_crio 6313
This theorem is referenced by:  dfoi  7242  oieq1  7243  oieq2  7244  ordtypecbv  7248  ordtypelem3  7251  zorn2lem1  8139  zorn2g  8146  cidfval  13594  cidval  13595  cidpropd  13629  lubfval  14128  glbfval  14133  spwval2  14349  spwval  14350  grpinvfval  14536  pj1fval  15019  mpfrcl  19418  evlsval  19419  q1pval  19555  ig1pval  19574  gidval  20896  grpoinvfval  20907  pjhfval  21991  cvmliftlem5  23835  cvmliftlem15  23844  rngounval2  25528  issubcv  25773  trlfset  30971  dicffval  31986  dicfval  31987  dihffval  32042  dihfval  32043  hvmapffval  32570  hvmapfval  32571  hdmap1fval  32609  hdmapffval  32641  hdmapfval  32642  hgmapfval  32701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-riota 6320
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