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Theorem unjust 3157
Description: Soundness justification theorem for df-un 3158. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
unjust  |-  { 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
Dummy variable  z is distinct from all other variables.

Proof of Theorem unjust
StepHypRef Expression
1 eleq1 2344 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 eleq1 2344 . . . 4  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
31, 2orbi12d 692 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  \/  x  e.  B
)  <->  ( z  e.  A  \/  z  e.  B ) ) )
43cbvabv 2403 . 2  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { z  |  ( z  e.  A  \/  z  e.  B ) }
5 eleq1 2344 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
6 eleq1 2344 . . . 4  |-  ( z  =  y  ->  (
z  e.  B  <->  y  e.  B ) )
75, 6orbi12d 692 . . 3  |-  ( z  =  y  ->  (
( z  e.  A  \/  z  e.  B
)  <->  ( y  e.  A  \/  y  e.  B ) ) )
87cbvabv 2403 . 2  |-  { z  |  ( z  e.  A  \/  z  e.  B ) }  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
94, 8eqtri 2304 1  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 359    = wceq 1624    e. wcel 1685   {cab 2270
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280
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