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Theorem riotabidva 6321
Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2779 analog.) (Contributed by NM, 17-Jan-2012.)
Hypothesis
Ref Expression
riotabidva.1  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
riotabidva  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem riotabidva
StepHypRef Expression
1 riotabidva.1 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
21pm5.32da 622 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
32iotabidv 5240 . . . . 5  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  A  /\  ch ) ) )
43adantr 451 . . . 4  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  A  /\  ch ) ) )
54ifeq1da 3590 . . 3  |-  ( ph  ->  if ( E! x  e.  A  ps ,  ( iota x ( x  e.  A  /\  ps ) ) ,  (
Undef `  { x  |  x  e.  A }
) )  =  if ( E! x  e.  A  ps ,  ( iota x ( x  e.  A  /\  ch ) ) ,  (
Undef `  { x  |  x  e.  A }
) ) )
61reubidva 2723 . . . 4  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ch )
)
76ifbid 3583 . . 3  |-  ( ph  ->  if ( E! x  e.  A  ps ,  ( iota x ( x  e.  A  /\  ch ) ) ,  (
Undef `  { x  |  x  e.  A }
) )  =  if ( E! x  e.  A  ch ,  ( iota x ( x  e.  A  /\  ch ) ) ,  (
Undef `  { x  |  x  e.  A }
) ) )
85, 7eqtrd 2315 . 2  |-  ( ph  ->  if ( E! x  e.  A  ps ,  ( iota x ( x  e.  A  /\  ps ) ) ,  (
Undef `  { x  |  x  e.  A }
) )  =  if ( E! x  e.  A  ch ,  ( iota x ( x  e.  A  /\  ch ) ) ,  (
Undef `  { x  |  x  e.  A }
) ) )
9 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 }
) )
10 df-riota 6304 . 2  |-  ( iota_ x  e.  A ch )  =  if ( E! x  e.  A  ch ,  ( iota x ( x  e.  A  /\  ch ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
118, 9, 103eqtr4g 2340 1  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E!wreu 2545   ifcif 3565   iotacio 5217   ` cfv 5255   Undefcund 6296   iota_crio 6297
This theorem is referenced by:  riotabiia  6322  cidpropd  13613  grpinvpropd  14543  grpoidval  20883  adjval2  22471  xdivval  23102  issubcv  25670  glbconN  29566  cdlemk33N  31098  cdlemk34  31099  cdlemkid4  31123
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-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-un 3157  df-if 3566  df-uni 3828  df-iota 5219  df-riota 6304
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