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Theorem riotaeqdv 6486
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 . . . . . . 7  |-  ( ph  ->  A  =  B )
21eleq2d 2454 . . . . . 6  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
32anbi1d 686 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ps ) ) )
43eubidv 2246 . . . 4  |-  ( ph  ->  ( E! x ( x  e.  A  /\  ps )  <->  E! x ( x  e.  B  /\  ps ) ) )
5 df-reu 2656 . . . 4  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
6 df-reu 2656 . . . 4  |-  ( E! x  e.  B  ps  <->  E! x ( x  e.  B  /\  ps )
)
74, 5, 63bitr4g 280 . . 3  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  B  ps )
)
83iotabidv 5379 . . 3  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  B  /\  ps ) ) )
92abbidv 2501 . . . 4  |-  ( ph  ->  { x  |  x  e.  A }  =  { x  |  x  e.  B } )
109fveq2d 5672 . . 3  |-  ( ph  ->  ( Undef `  { x  |  x  e.  A } )  =  (
Undef `  { x  |  x  e.  B }
) )
117, 8, 10ifbieq12d 3704 . 2  |-  ( ph  ->  if ( E! x  e.  A  ps ,  ( iota x ( x  e.  A  /\  ps ) ) ,  (
Undef `  { x  |  x  e.  A }
) )  =  if ( E! x  e.  B  ps ,  ( iota x ( x  e.  B  /\  ps ) ) ,  (
Undef `  { x  |  x  e.  B }
) ) )
12 df-riota 6485 . 2  |-  ( iota_ x  e.  A ps )  =  if ( E! x  e.  A  ps ,  ( iota x ( x  e.  A  /\  ps ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
13 df-riota 6485 . 2  |-  ( iota_ x  e.  B ps )  =  if ( E! x  e.  B  ps ,  ( iota x ( x  e.  B  /\  ps ) ) ,  (
Undef `  { x  |  x  e.  B }
) )
1411, 12, 133eqtr4g 2444 1  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  B ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   E!weu 2238   {cab 2373   E!wreu 2651   ifcif 3682   iotacio 5356   ` cfv 5394   Undefcund 6477   iota_crio 6478
This theorem is referenced by:  riotaeqbidv  6488  grpinvpropd  14793  funtransport  25679  fvtransport  25680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-riota 6485
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