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Theorem raldifb 3138
Description: Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
Assertion
Ref Expression
raldifb  |-  ( A. x  e.  A  (
x  e/  B  ->  ph )  <->  A. x  e.  ( A  \  B )
ph )

Proof of Theorem raldifb
StepHypRef Expression
1 impexp 259 . . . 4  |-  ( ( ( x  e.  A  /\  x  e/  B )  ->  ph )  <->  ( x  e.  A  ->  ( x  e/  B  ->  ph )
) )
21bicomi 130 . . 3  |-  ( ( x  e.  A  -> 
( x  e/  B  ->  ph ) )  <->  ( (
x  e.  A  /\  x  e/  B )  ->  ph ) )
3 df-nel 2351 . . . . . 6  |-  ( x  e/  B  <->  -.  x  e.  B )
43anbi2i 445 . . . . 5  |-  ( ( x  e.  A  /\  x  e/  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3006 . . . . . 6  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
65bicomi 130 . . . . 5  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  ( A  \  B ) )
74, 6bitri 182 . . . 4  |-  ( ( x  e.  A  /\  x  e/  B )  <->  x  e.  ( A  \  B ) )
87imbi1i 236 . . 3  |-  ( ( ( x  e.  A  /\  x  e/  B )  ->  ph )  <->  ( x  e.  ( A  \  B
)  ->  ph ) )
92, 8bitri 182 . 2  |-  ( ( x  e.  A  -> 
( x  e/  B  ->  ph ) )  <->  ( x  e.  ( A  \  B
)  ->  ph ) )
109ralbii2 2388 1  |-  ( A. x  e.  A  (
x  e/  B  ->  ph )  <->  A. x  e.  ( A  \  B )
ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438    e/ wnel 2350   A.wral 2359    \ cdif 2994
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-nel 2351  df-ral 2364  df-v 2621  df-dif 2999
This theorem is referenced by: (None)
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