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Theorem riotaeqdv 5699
Description: Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
Hypothesis
Ref Expression
riotaeqdv.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
riotaeqdv  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  B  ps ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)    B( x)

Proof of Theorem riotaeqdv
StepHypRef Expression
1 riotaeqdv.1 . . . . 5  |-  ( ph  ->  A  =  B )
21eleq2d 2187 . . . 4  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
32anbi1d 460 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ps ) ) )
43iotabidv 5079 . 2  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  B  /\  ps ) ) )
5 df-riota 5698 . 2  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
6 df-riota 5698 . 2  |-  ( iota_ x  e.  B  ps )  =  ( iota x
( x  e.  B  /\  ps ) )
74, 5, 63eqtr4g 2175 1  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   iotacio 5056   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:  riotaeqbidv  5701
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