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Theorem difjust 2998
Description: Soundness justification theorem for df-dif 2999. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
difjust  |-  { x  |  ( x  e.  A  /\  -.  x  e.  B ) }  =  { y  |  ( y  e.  A  /\  -.  y  e.  B
) }
Distinct variable groups:    x, A    x, B    y, A    y, B

Proof of Theorem difjust
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eleq1 2150 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 eleq1 2150 . . . . 5  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
32notbid 627 . . . 4  |-  ( x  =  z  ->  ( -.  x  e.  B  <->  -.  z  e.  B ) )
41, 3anbi12d 457 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  /\  -.  x  e.  B
)  <->  ( z  e.  A  /\  -.  z  e.  B ) ) )
54cbvabv 2211 . 2  |-  { x  |  ( x  e.  A  /\  -.  x  e.  B ) }  =  { z  |  ( z  e.  A  /\  -.  z  e.  B
) }
6 eleq1 2150 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
7 eleq1 2150 . . . . 5  |-  ( z  =  y  ->  (
z  e.  B  <->  y  e.  B ) )
87notbid 627 . . . 4  |-  ( z  =  y  ->  ( -.  z  e.  B  <->  -.  y  e.  B ) )
96, 8anbi12d 457 . . 3  |-  ( z  =  y  ->  (
( z  e.  A  /\  -.  z  e.  B
)  <->  ( y  e.  A  /\  -.  y  e.  B ) ) )
109cbvabv 2211 . 2  |-  { z  |  ( z  e.  A  /\  -.  z  e.  B ) }  =  { y  |  ( y  e.  A  /\  -.  y  e.  B
) }
115, 10eqtri 2108 1  |-  { x  |  ( x  e.  A  /\  -.  x  e.  B ) }  =  { y  |  ( y  e.  A  /\  -.  y  e.  B
) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074
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-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084
This theorem is referenced by: (None)
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