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Theorem disjeq2dv 4129
Description: Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypothesis
Ref Expression
disjeq2dv.1  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
Assertion
Ref Expression
disjeq2dv  |-  ( ph  ->  (Disj  x  e.  A B 
<-> Disj  x  e.  A C
) )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem disjeq2dv
StepHypRef Expression
1 disjeq2dv.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
21ralrimiva 2733 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 disjeq2 4128 . 2  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A B  <-> Disj  x  e.  A C ) )
42, 3syl 16 1  |-  ( ph  ->  (Disj  x  e.  A B 
<-> Disj  x  e.  A C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650  Disj wdisj 4124
This theorem is referenced by:  disjeq12d  4133  iunmbl  19315  uniioovol  19339  voliunnfl  25956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-ral 2655  df-rmo 2658  df-in 3271  df-ss 3278  df-disj 4125
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