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Theorem raldifb 3447
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 434 . . . 4  |-  ( ( ( x  e.  A  /\  x  e/  B )  ->  ph )  <->  ( x  e.  A  ->  ( x  e/  B  ->  ph )
) )
21bicomi 194 . . 3  |-  ( ( x  e.  A  -> 
( x  e/  B  ->  ph ) )  <->  ( (
x  e.  A  /\  x  e/  B )  ->  ph ) )
3 df-nel 2570 . . . . . 6  |-  ( x  e/  B  <->  -.  x  e.  B )
43anbi2i 676 . . . . 5  |-  ( ( x  e.  A  /\  x  e/  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3290 . . . . . 6  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
65bicomi 194 . . . . 5  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  ( A  \  B ) )
74, 6bitri 241 . . . 4  |-  ( ( x  e.  A  /\  x  e/  B )  <->  x  e.  ( A  \  B ) )
87imbi1i 316 . . 3  |-  ( ( ( x  e.  A  /\  x  e/  B )  ->  ph )  <->  ( x  e.  ( A  \  B
)  ->  ph ) )
92, 8bitri 241 . 2  |-  ( ( x  e.  A  -> 
( x  e/  B  ->  ph ) )  <->  ( x  e.  ( A  \  B
)  ->  ph ) )
109ralbii2 2694 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    <-> wb 177    /\ wa 359    e. wcel 1721    e/ wnel 2568   A.wral 2666    \ cdif 3277
This theorem is referenced by:  cusgrares  21434  raldifsnb  27946  2spotdisj  28164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-nel 2570  df-ral 2671  df-v 2918  df-dif 3283
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