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Theorem difjust 3324
Description: Soundness justification theorem for df-dif 3325. (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 2498 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 eleq1 2498 . . . . 5  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
32notbid 287 . . . 4  |-  ( x  =  z  ->  ( -.  x  e.  B  <->  -.  z  e.  B ) )
41, 3anbi12d 693 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  /\  -.  x  e.  B
)  <->  ( z  e.  A  /\  -.  z  e.  B ) ) )
54cbvabv 2557 . 2  |-  { x  |  ( x  e.  A  /\  -.  x  e.  B ) }  =  { z  |  ( z  e.  A  /\  -.  z  e.  B
) }
6 eleq1 2498 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
7 eleq1 2498 . . . . 5  |-  ( z  =  y  ->  (
z  e.  B  <->  y  e.  B ) )
87notbid 287 . . . 4  |-  ( z  =  y  ->  ( -.  z  e.  B  <->  -.  y  e.  B ) )
96, 8anbi12d 693 . . 3  |-  ( z  =  y  ->  (
( z  e.  A  /\  -.  z  e.  B
)  <->  ( y  e.  A  /\  -.  y  e.  B ) ) )
109cbvabv 2557 . 2  |-  { z  |  ( z  e.  A  /\  -.  z  e.  B ) }  =  { y  |  ( y  e.  A  /\  -.  y  e.  B
) }
115, 10eqtri 2458 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 360    = wceq 1653    e. wcel 1726   {cab 2424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434
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