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Theorem disjeq1d 3884
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypothesis
Ref Expression
disjeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
disjeq1d  |-  ( ph  ->  (Disj  x  e.  A  C 
<-> Disj  x  e.  B  C
) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    C( x)

Proof of Theorem disjeq1d
StepHypRef Expression
1 disjeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 disjeq1 3883 . 2  |-  ( A  =  B  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  C ) )
31, 2syl 14 1  |-  ( ph  ->  (Disj  x  e.  A  C 
<-> Disj  x  e.  B  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316  Disj wdisj 3876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-rmo 2401  df-in 3047  df-ss 3054  df-disj 3877
This theorem is referenced by:  disjeq12d  3885
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