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Theorem disjeq2 3997
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 3231 . . . 4  |-  ( B  =  C  ->  C  C_  B )
21ralimi 2618 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  C  C_  B
)
3 disjss2 3996 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  (Disj  x  e.  A B  -> Disj  x  e.  A C ) )
42, 3syl 15 . 2  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A B  -> Disj  x  e.  A C ) )
5 eqimss 3230 . . . 4  |-  ( B  =  C  ->  B  C_  C )
65ralimi 2618 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  B  C_  C
)
7 disjss2 3996 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )
86, 7syl 15 . 2  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )
94, 8impbid 183 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 176    = wceq 1623   A.wral 2543    C_ wss 3152  Disj wdisj 3993
This theorem is referenced by:  disjeq2dv  3998  voliun  18911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-rmo 2551  df-in 3159  df-ss 3166  df-disj 3994
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