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Theorem disjeq2dv 4412
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 2857 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 disjeq2 4411 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793  Disj wdisj 4407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-ral 2798  df-rmo 2801  df-in 3468  df-ss 3475  df-disj 4408
This theorem is referenced by:  disjeq12d  4416  iunmbl  21836  uniioovol  21861  voliunnfl  30033
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