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Theorem raldifb 3585
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 446 . . . 4  |-  ( ( ( x  e.  A  /\  x  e/  B )  ->  ph )  <->  ( x  e.  A  ->  ( x  e/  B  ->  ph )
) )
21bicomi 204 . . 3  |-  ( ( x  e.  A  -> 
( x  e/  B  ->  ph ) )  <->  ( (
x  e.  A  /\  x  e/  B )  ->  ph ) )
3 df-nel 2603 . . . . . 6  |-  ( x  e/  B  <->  -.  x  e.  B )
43anbi2i 694 . . . . 5  |-  ( ( x  e.  A  /\  x  e/  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3426 . . . . . 6  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
65bicomi 204 . . . . 5  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  ( A  \  B ) )
74, 6bitri 251 . . . 4  |-  ( ( x  e.  A  /\  x  e/  B )  <->  x  e.  ( A  \  B ) )
87imbi1i 325 . . 3  |-  ( ( ( x  e.  A  /\  x  e/  B )  ->  ph )  <->  ( x  e.  ( A  \  B
)  ->  ph ) )
92, 8bitri 251 . 2  |-  ( ( x  e.  A  -> 
( x  e/  B  ->  ph ) )  <->  ( x  e.  ( A  \  B
)  ->  ph ) )
109ralbii2 2835 1  |-  ( A. x  e.  A  (
x  e/  B  ->  ph )  <->  A. x  e.  ( A  \  B )
ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369    e. wcel 1844    e/ wnel 2601   A.wral 2756    \ cdif 3413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-nel 2603  df-ral 2761  df-v 3063  df-dif 3419
This theorem is referenced by:  raldifsnb  4105  cusgrares  24901  2spotdisj  25490  aacllem  38873
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