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Theorem injust 2951
Description: Soundness justification theorem for df-in 2952. (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 2116 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 eleq1 2116 . . . 4  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
31, 2anbi12d 450 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  /\  x  e.  B
)  <->  ( z  e.  A  /\  z  e.  B ) ) )
43cbvabv 2177 . 2  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { z  |  ( z  e.  A  /\  z  e.  B ) }
5 eleq1 2116 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
6 eleq1 2116 . . . 4  |-  ( z  =  y  ->  (
z  e.  B  <->  y  e.  B ) )
75, 6anbi12d 450 . . 3  |-  ( z  =  y  ->  (
( z  e.  A  /\  z  e.  B
)  <->  ( y  e.  A  /\  y  e.  B ) ) )
87cbvabv 2177 . 2  |-  { z  |  ( z  e.  A  /\  z  e.  B ) }  =  { y  |  ( y  e.  A  /\  y  e.  B ) }
94, 8eqtri 2076 1  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { y  |  ( y  e.  A  /\  y  e.  B ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259    e. wcel 1409   {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052
This theorem is referenced by: (None)
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