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Theorem difeqri2 25143
Description: Inference from membership to difference. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
difeqri2  |-  ( A. x ( ( x  e.  A  /\  -.  x  e.  B )  <->  x  e.  C )  -> 
( A  \  B
)  =  C )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem difeqri2
StepHypRef Expression
1 eldif 3175 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
21bibi1i 305 . . 3  |-  ( ( x  e.  ( A 
\  B )  <->  x  e.  C )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  C
) )
32albii 1556 . 2  |-  ( A. x ( x  e.  ( A  \  B
)  <->  x  e.  C
)  <->  A. x ( ( x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  C
) )
4 dfcleq 2290 . . 3  |-  ( ( A  \  B )  =  C  <->  A. x
( x  e.  ( A  \  B )  <-> 
x  e.  C ) )
54biimpri 197 . 2  |-  ( A. x ( x  e.  ( A  \  B
)  <->  x  e.  C
)  ->  ( A  \  B )  =  C )
63, 5sylbir 204 1  |-  ( A. x ( ( x  e.  A  /\  -.  x  e.  B )  <->  x  e.  C )  -> 
( A  \  B
)  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696    \ cdif 3162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168
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