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Theorem unjust 3324
Description: Soundness justification theorem for df-un 3325. (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

Proof of Theorem unjust
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eleq1 2496 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 eleq1 2496 . . . 4  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
31, 2orbi12d 691 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  \/  x  e.  B
)  <->  ( z  e.  A  \/  z  e.  B ) ) )
43cbvabv 2555 . 2  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { z  |  ( z  e.  A  \/  z  e.  B ) }
5 eleq1 2496 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
6 eleq1 2496 . . . 4  |-  ( z  =  y  ->  (
z  e.  B  <->  y  e.  B ) )
75, 6orbi12d 691 . . 3  |-  ( z  =  y  ->  (
( z  e.  A  \/  z  e.  B
)  <->  ( y  e.  A  \/  y  e.  B ) ) )
87cbvabv 2555 . 2  |-  { z  |  ( z  e.  A  \/  z  e.  B ) }  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
94, 8eqtri 2456 1  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1652    e. wcel 1725   {cab 2422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432
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