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Theorem disjeq2dv 4179
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 2781 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 disjeq2 4178 . 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 1652    e. wcel 1725   A.wral 2697  Disj wdisj 4174
This theorem is referenced by:  disjeq12d  4183  iunmbl  19439  uniioovol  19463  voliunnfl  26240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-ral 2702  df-rmo 2705  df-in 3319  df-ss 3326  df-disj 4175
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