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Theorem rabbi2dva 3145
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 2925 . 2  |-  ( A  i^i  B )  =  { x  e.  A  |  x  e.  B }
2 rabbi2dva.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  B  <->  ps )
)
32rabbidva 2548 . 2  |-  ( ph  ->  { x  e.  A  |  x  e.  B }  =  { x  e.  A  |  ps } )
41, 3syl5eq 2084 1  |-  ( ph  ->  ( A  i^i  B
)  =  { x  e.  A  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   {crab 2310    i^i cin 2916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-ral 2311  df-rab 2315  df-in 2924
This theorem is referenced by:  fndmdif  5272
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