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Theorem disjeq2dv 3998
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 2626 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 disjeq2 3997 . 2  |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A B  <-> Disj  x  e.  A C ) )
42, 3syl 15 1  |-  ( ph  ->  (Disj  x  e.  A B 
<-> Disj  x  e.  A C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543  Disj wdisj 3993
This theorem is referenced by:  disjeq12d  4002  iunmbl  18910  uniioovol  18934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-rmo 2551  df-in 3159  df-ss 3166  df-disj 3994
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