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Theorem disjeq1d 3914
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 3913 . 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 1331  Disj wdisj 3906
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-rmo 2424  df-in 3077  df-ss 3084  df-disj 3907
This theorem is referenced by:  disjeq12d  3915
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