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Theorem disjeq1 3752
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjeq1  |-  ( A  =  B  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  C ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem disjeq1
StepHypRef Expression
1 eqimss2 2998 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 disjss1 3751 . . 3  |-  ( B 
C_  A  ->  (Disj  x  e.  A  C  -> Disj  x  e.  B  C ) )
31, 2syl 14 . 2  |-  ( A  =  B  ->  (Disj  x  e.  A  C  -> Disj  x  e.  B  C ) )
4 eqimss 2997 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 disjss1 3751 . . 3  |-  ( A 
C_  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
64, 5syl 14 . 2  |-  ( A  =  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
73, 6impbid 120 1  |-  ( A  =  B  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243    C_ wss 2917  Disj wdisj 3745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-rmo 2314  df-in 2924  df-ss 2931  df-disj 3746
This theorem is referenced by:  disjeq1d  3753
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