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Theorem disjeq12d 4002
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 4001 . 2  |-  ( ph  ->  (Disj  x  e.  A C 
<-> Disj  x  e.  B C
) )
3 disjeq12d.1 . . . 4  |-  ( ph  ->  C  =  D )
43adantr 451 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  C  =  D )
54disjeq2dv 3998 . 2  |-  ( ph  ->  (Disj  x  e.  B C 
<-> Disj  x  e.  B D
) )
62, 5bitrd 244 1  |-  ( ph  ->  (Disj  x  e.  A C 
<-> Disj  x  e.  B D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684  Disj wdisj 3993
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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