MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjeq2dv Structured version   Unicode version

Theorem disjeq2dv 4368
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 2825 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
3 disjeq2 4367 . 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 1370    e. wcel 1758   A.wral 2795  Disj wdisj 4363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-ral 2800  df-rmo 2803  df-in 3436  df-ss 3443  df-disj 4364
This theorem is referenced by:  disjeq12d  4372  iunmbl  21160  uniioovol  21185  voliunnfl  28576
  Copyright terms: Public domain W3C validator