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Theorem rabbi2dva 3197
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 2995 . 2  |-  ( A  i^i  B )  =  { x  e.  A  |  x  e.  B }
2 rabbi2dva.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  B  <->  ps )
)
32rabbidva 2603 . 2  |-  ( ph  ->  { x  e.  A  |  x  e.  B }  =  { x  e.  A  |  ps } )
41, 3syl5eq 2129 1  |-  ( ph  ->  ( A  i^i  B
)  =  { x  e.  A  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   {crab 2359    i^i cin 2987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-ral 2360  df-rab 2364  df-in 2994
This theorem is referenced by:  fndmdif  5361
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