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Theorem riotaeqdv 6305
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 2350 . . . . . 6  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
32anbi1d 685 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ps ) ) )
43eubidv 2151 . . . 4  |-  ( ph  ->  ( E! x ( x  e.  A  /\  ps )  <->  E! x ( x  e.  B  /\  ps ) ) )
5 df-reu 2550 . . . 4  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
6 df-reu 2550 . . . 4  |-  ( E! x  e.  B  ps  <->  E! x ( x  e.  B  /\  ps )
)
74, 5, 63bitr4g 279 . . 3  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  B  ps )
)
83iotabidv 5240 . . 3  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  B  /\  ps ) ) )
92abbidv 2397 . . . 4  |-  ( ph  ->  { x  |  x  e.  A }  =  { x  |  x  e.  B } )
109fveq2d 5529 . . 3  |-  ( ph  ->  ( Undef `  { x  |  x  e.  A } )  =  (
Undef `  { x  |  x  e.  B }
) )
117, 8, 10ifbieq12d 3587 . 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 6304 . 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 6304 . 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 2340 1  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  B ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269   E!wreu 2545   ifcif 3565   iotacio 5217   ` cfv 5255   Undefcund 6296   iota_crio 6297
This theorem is referenced by:  riotaeqbidv  6307  grpinvpropd  14543  funtransport  24654  fvtransport  24655  linevala2  26078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-riota 6304
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