ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  disjeq12d Unicode version

Theorem disjeq12d 3973
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 3972 . 2  |-  ( ph  ->  (Disj  x  e.  A  C 
<-> Disj  x  e.  B  C
) )
3 disjeq12d.1 . . . 4  |-  ( ph  ->  C  =  D )
43adantr 274 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  C  =  D )
54disjeq2dv 3969 . 2  |-  ( ph  ->  (Disj  x  e.  B  C 
<-> Disj  x  e.  B  D
) )
62, 5bitrd 187 1  |-  ( ph  ->  (Disj  x  e.  A  C 
<-> Disj  x  e.  B  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348    e. wcel 2141  Disj wdisj 3964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-rmo 2456  df-in 3127  df-ss 3134  df-disj 3965
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator