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Theorem riotaeqdv 6553
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 2505 . . . . . 6  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
32anbi1d 687 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ps ) ) )
43eubidv 2291 . . . 4  |-  ( ph  ->  ( E! x ( x  e.  A  /\  ps )  <->  E! x ( x  e.  B  /\  ps ) ) )
5 df-reu 2714 . . . 4  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
6 df-reu 2714 . . . 4  |-  ( E! x  e.  B  ps  <->  E! x ( x  e.  B  /\  ps )
)
74, 5, 63bitr4g 281 . . 3  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  B  ps )
)
83iotabidv 5442 . . 3  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  B  /\  ps ) ) )
92abbidv 2552 . . . 4  |-  ( ph  ->  { x  |  x  e.  A }  =  { x  |  x  e.  B } )
109fveq2d 5735 . . 3  |-  ( ph  ->  ( Undef `  { x  |  x  e.  A } )  =  (
Undef `  { x  |  x  e.  B }
) )
117, 8, 10ifbieq12d 3763 . 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 6552 . 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 6552 . 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 2495 1  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  B ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E!weu 2283   {cab 2424   E!wreu 2709   ifcif 3741   iotacio 5419   ` cfv 5457   Undefcund 6544   iota_crio 6545
This theorem is referenced by:  riotaeqbidv  6555  grpinvpropd  14871  funtransport  25970  fvtransport  25971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-riota 6552
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