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Theorem disjeq2 4188
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 3403 . . . 4  |-  ( B  =  C  ->  C  C_  B )
21ralimi 2783 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  C  C_  B
)
3 disjss2 4187 . . 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 3402 . . . 4  |-  ( B  =  C  ->  B  C_  C )
65ralimi 2783 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  B  C_  C
)
7 disjss2 4187 . . 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 185 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 178    = wceq 1653   A.wral 2707    C_ wss 3322  Disj wdisj 4184
This theorem is referenced by:  disjeq2dv  4189  voliun  19450  mblfinlem2  26246  voliunnfl  26252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-ral 2712  df-rmo 2715  df-in 3329  df-ss 3336  df-disj 4185
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