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Theorem disjeq12d 4191
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disjeq1d.1  |-  ( ph  ->  A  =  B )
disjeq12d.1  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
disjeq12d  |-  ( ph  ->  (Disj  x  e.  A C 
<-> Disj  x  e.  B D
) )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    C( x)    D( x)

Proof of Theorem disjeq12d
StepHypRef Expression
1 disjeq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21disjeq1d 4190 . 2  |-  ( ph  ->  (Disj  x  e.  A C 
<-> Disj  x  e.  B C
) )
3 disjeq12d.1 . . . 4  |-  ( ph  ->  C  =  D )
43adantr 452 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  C  =  D )
54disjeq2dv 4187 . 2  |-  ( ph  ->  (Disj  x  e.  B C 
<-> Disj  x  e.  B D
) )
62, 5bitrd 245 1  |-  ( ph  ->  (Disj  x  e.  A C 
<-> Disj  x  e.  B D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725  Disj wdisj 4182
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 2417
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-ral 2710  df-rmo 2713  df-in 3327  df-ss 3334  df-disj 4183
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