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Theorem disjeq2 3984
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjeq2  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A  B  <-> Disj  x  e.  A  C ) )

Proof of Theorem disjeq2
StepHypRef Expression
1 eqimss2 3210 . . . 4  |-  ( B  =  C  ->  C  C_  B )
21ralimi 2540 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  C  C_  B
)
3 disjss2 3983 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  (Disj  x  e.  A  B  -> Disj  x  e.  A  C ) )
42, 3syl 14 . 2  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A  B  -> Disj  x  e.  A  C )
)
5 eqimss 3209 . . . 4  |-  ( B  =  C  ->  B  C_  C )
65ralimi 2540 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  B  C_  C
)
7 disjss2 3983 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B ) )
86, 7syl 14 . 2  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B )
)
94, 8impbid 129 1  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A  B  <-> Disj  x  e.  A  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353   A.wral 2455    C_ wss 3129  Disj wdisj 3980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-rmo 2463  df-in 3135  df-ss 3142  df-disj 3981
This theorem is referenced by:  disjeq2dv  3985
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