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Theorem unjust 2921
Description: Soundness justification theorem for df-un 2922. (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 2100 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 eleq1 2100 . . . 4  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
31, 2orbi12d 707 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  \/  x  e.  B
)  <->  ( z  e.  A  \/  z  e.  B ) ) )
43cbvabv 2161 . 2  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { z  |  ( z  e.  A  \/  z  e.  B ) }
5 eleq1 2100 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
6 eleq1 2100 . . . 4  |-  ( z  =  y  ->  (
z  e.  B  <->  y  e.  B ) )
75, 6orbi12d 707 . . 3  |-  ( z  =  y  ->  (
( z  e.  A  \/  z  e.  B
)  <->  ( y  e.  A  \/  y  e.  B ) ) )
87cbvabv 2161 . 2  |-  { z  |  ( z  e.  A  \/  z  e.  B ) }  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
94, 8eqtri 2060 1  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 629    = wceq 1243    e. wcel 1393   {cab 2026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036
This theorem is referenced by: (None)
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