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Theorem disjeq2 3790
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 3061 . . . 4  |-  ( B  =  C  ->  C  C_  B )
21ralimi 2431 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  C  C_  B
)
3 disjss2 3789 . . 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 3060 . . . 4  |-  ( B  =  C  ->  B  C_  C )
65ralimi 2431 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  B  C_  C
)
7 disjss2 3789 . . 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 127 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 103    = wceq 1285   A.wral 2353    C_ wss 2982  Disj wdisj 3786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-rmo 2361  df-in 2988  df-ss 2995  df-disj 3787
This theorem is referenced by:  disjeq2dv  3791
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