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Theorem raldifb 3216
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 261 . . . 4  |-  ( ( ( x  e.  A  /\  x  e/  B )  ->  ph )  <->  ( x  e.  A  ->  ( x  e/  B  ->  ph )
) )
21bicomi 131 . . 3  |-  ( ( x  e.  A  -> 
( x  e/  B  ->  ph ) )  <->  ( (
x  e.  A  /\  x  e/  B )  ->  ph ) )
3 df-nel 2404 . . . . . 6  |-  ( x  e/  B  <->  -.  x  e.  B )
43anbi2i 452 . . . . 5  |-  ( ( x  e.  A  /\  x  e/  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3080 . . . . . 6  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
65bicomi 131 . . . . 5  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  ( A  \  B ) )
74, 6bitri 183 . . . 4  |-  ( ( x  e.  A  /\  x  e/  B )  <->  x  e.  ( A  \  B ) )
87imbi1i 237 . . 3  |-  ( ( ( x  e.  A  /\  x  e/  B )  ->  ph )  <->  ( x  e.  ( A  \  B
)  ->  ph ) )
92, 8bitri 183 . 2  |-  ( ( x  e.  A  -> 
( x  e/  B  ->  ph ) )  <->  ( x  e.  ( A  \  B
)  ->  ph ) )
109ralbii2 2445 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 103    <-> wb 104    e. wcel 1480    e/ wnel 2403   A.wral 2416    \ cdif 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-nel 2404  df-ral 2421  df-v 2688  df-dif 3073
This theorem is referenced by: (None)
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