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Theorem rabbi2dva 3377
Description: Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
Hypothesis
Ref Expression
rabbi2dva.1  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  B  <->  ps )
)
Assertion
Ref Expression
rabbi2dva  |-  ( ph  ->  ( A  i^i  B
)  =  { x  e.  A  |  ps } )
Distinct variable groups:    ph, x    x, A    x, B
Allowed substitution hint:    ps( x)

Proof of Theorem rabbi2dva
StepHypRef Expression
1 dfin5 3160 . 2  |-  ( A  i^i  B )  =  { x  e.  A  |  x  e.  B }
2 rabbi2dva.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  B  <->  ps )
)
32rabbidva 2779 . 2  |-  ( ph  ->  { x  e.  A  |  x  e.  B }  =  { x  e.  A  |  ps } )
41, 3syl5eq 2327 1  |-  ( ph  ->  ( A  i^i  B
)  =  { x  e.  A  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    i^i cin 3151
This theorem is referenced by:  fndmdif  5629  bitsshft  12666  sylow3lem2  14939  leordtvallem1  16940  leordtvallem2  16941  ordtt1  17107  xkoccn  17313  txcnmpt  17318  xkopt  17349  ordthmeolem  17492  divstgphaus  17805  itg2monolem1  19105  lhop1  19361  efopn  20005  dirith  20678  pjvec  22275  pjocvec  22276  neibastop3  26311  diarnN  31319
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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-ral 2548  df-rab 2552  df-in 3159
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