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Theorem disjeq2 4431
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 3552 . . . 4  |-  ( B  =  C  ->  C  C_  B )
21ralimi 2850 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  C  C_  B
)
3 disjss2 4430 . . 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 3551 . . . 4  |-  ( B  =  C  ->  B  C_  C )
65ralimi 2850 . . 3  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  B  C_  C
)
7 disjss2 4430 . . 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 1395   A.wral 2807    C_ wss 3471  Disj wdisj 4427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-ral 2812  df-rmo 2815  df-in 3478  df-ss 3485  df-disj 4428
This theorem is referenced by:  disjeq2dv  4432  voliun  22090  carsgclctunlem2  28461  mblfinlem2  30236  voliunnfl  30242
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