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Theorem disjeq2 4421
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 3557 . . . 4  |-  ( B  =  C  ->  C  C_  B )
21ralimi 2857 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  C  C_  B
)
3 disjss2 4420 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  (Disj  x  e.  A  B  -> Disj  x  e.  A  C ) )
42, 3syl 16 . 2  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A  B  -> Disj  x  e.  A  C )
)
5 eqimss 3556 . . . 4  |-  ( B  =  C  ->  B  C_  C )
65ralimi 2857 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  B  C_  C
)
7 disjss2 4420 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B ) )
86, 7syl 16 . 2  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B )
)
94, 8impbid 191 1  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A  B  <-> Disj  x  e.  A  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   A.wral 2814    C_ wss 3476  Disj wdisj 4417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-ral 2819  df-rmo 2822  df-in 3483  df-ss 3490  df-disj 4418
This theorem is referenced by:  disjeq2dv  4422  voliun  21727  mblfinlem2  29657  voliunnfl  29663
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