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Theorem injust 3046
Description: Soundness justification theorem for df-in 3047. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
injust  |-  { 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 injust
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eleq1 2180 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 eleq1 2180 . . . 4  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
31, 2anbi12d 464 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  /\  x  e.  B
)  <->  ( z  e.  A  /\  z  e.  B ) ) )
43cbvabv 2241 . 2  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { z  |  ( z  e.  A  /\  z  e.  B ) }
5 eleq1 2180 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
6 eleq1 2180 . . . 4  |-  ( z  =  y  ->  (
z  e.  B  <->  y  e.  B ) )
75, 6anbi12d 464 . . 3  |-  ( z  =  y  ->  (
( z  e.  A  /\  z  e.  B
)  <->  ( y  e.  A  /\  y  e.  B ) ) )
87cbvabv 2241 . 2  |-  { z  |  ( z  e.  A  /\  z  e.  B ) }  =  { y  |  ( y  e.  A  /\  y  e.  B ) }
94, 8eqtri 2138 1  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { y  |  ( y  e.  A  /\  y  e.  B ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    e. wcel 1465   {cab 2103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113
This theorem is referenced by: (None)
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