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Theorem difjust 3154
Description: Soundness justification theorem for df-dif 3155. (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 2343 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 eleq1 2343 . . . . 5  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
32notbid 285 . . . 4  |-  ( x  =  z  ->  ( -.  x  e.  B  <->  -.  z  e.  B ) )
41, 3anbi12d 691 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  /\  -.  x  e.  B
)  <->  ( z  e.  A  /\  -.  z  e.  B ) ) )
54cbvabv 2402 . 2  |-  { x  |  ( x  e.  A  /\  -.  x  e.  B ) }  =  { z  |  ( z  e.  A  /\  -.  z  e.  B
) }
6 eleq1 2343 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
7 eleq1 2343 . . . . 5  |-  ( z  =  y  ->  (
z  e.  B  <->  y  e.  B ) )
87notbid 285 . . . 4  |-  ( z  =  y  ->  ( -.  z  e.  B  <->  -.  y  e.  B ) )
96, 8anbi12d 691 . . 3  |-  ( z  =  y  ->  (
( z  e.  A  /\  -.  z  e.  B
)  <->  ( y  e.  A  /\  -.  y  e.  B ) ) )
109cbvabv 2402 . 2  |-  { z  |  ( z  e.  A  /\  -.  z  e.  B ) }  =  { y  |  ( y  e.  A  /\  -.  y  e.  B
) }
115, 10eqtri 2303 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 358    = wceq 1623    e. wcel 1684   {cab 2269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279
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