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Theorem riotaeqdv 6542
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 2502 . . . . . 6  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
32anbi1d 686 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ps ) ) )
43eubidv 2288 . . . 4  |-  ( ph  ->  ( E! x ( x  e.  A  /\  ps )  <->  E! x ( x  e.  B  /\  ps ) ) )
5 df-reu 2704 . . . 4  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
6 df-reu 2704 . . . 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 5431 . . 3  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  B  /\  ps ) ) )
92abbidv 2549 . . . 4  |-  ( ph  ->  { x  |  x  e.  A }  =  { x  |  x  e.  B } )
109fveq2d 5724 . . 3  |-  ( ph  ->  ( Undef `  { x  |  x  e.  A } )  =  (
Undef `  { x  |  x  e.  B }
) )
117, 8, 10ifbieq12d 3753 . 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 6541 . 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 6541 . 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 2492 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 1652    e. wcel 1725   E!weu 2280   {cab 2421   E!wreu 2699   ifcif 3731   iotacio 5408   ` cfv 5446   Undefcund 6533   iota_crio 6534
This theorem is referenced by:  riotaeqbidv  6544  grpinvpropd  14858  funtransport  25957  fvtransport  25958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-riota 6541
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