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Theorem riotaeqdv 5497
Description: Formula-building deduction rule 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 2123 . . . 4  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
32anbi1d 446 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ps ) ) )
43iotabidv 4916 . 2  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  B  /\  ps ) ) )
5 df-riota 5496 . 2  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
6 df-riota 5496 . 2  |-  ( iota_ x  e.  B  ps )  =  ( iota x
( x  e.  B  /\  ps ) )
74, 5, 63eqtr4g 2113 1  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   iotacio 4893   iota_crio 5495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-uni 3609  df-iota 4895  df-riota 5496
This theorem is referenced by:  riotaeqbidv  5499
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