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Theorem riotaeqbidv 5701
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 5700 . 2  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  A  ch ) )
3 riotaeqbidv.1 . . 3  |-  ( ph  ->  A  =  B )
43riotaeqdv 5699 . 2  |-  ( ph  ->  ( iota_ x  e.  A  ch )  =  ( iota_ x  e.  B  ch ) )
52, 4eqtrd 2150 1  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  B  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316   iota_crio 5697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-uni 3707  df-iota 5058  df-riota 5698
This theorem is referenced by:  acexmidlemab  5736
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