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Theorem cbvdisj 3832
Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
cbvdisj.1  |-  F/_ y B
cbvdisj.2  |-  F/_ x C
cbvdisj.3  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbvdisj  |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  C )
Distinct variable group:    x, y, A
Allowed substitution hints:    B( x, y)    C( x, y)

Proof of Theorem cbvdisj
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvdisj.1 . . . . 5  |-  F/_ y B
21nfcri 2222 . . . 4  |-  F/ y  z  e.  B
3 cbvdisj.2 . . . . 5  |-  F/_ x C
43nfcri 2222 . . . 4  |-  F/ x  z  e.  C
5 cbvdisj.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2157 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvrmo 2589 . . 3  |-  ( E* x  e.  A  z  e.  B  <->  E* y  e.  A  z  e.  C )
87albii 1404 . 2  |-  ( A. z E* x  e.  A  z  e.  B  <->  A. z E* y  e.  A  z  e.  C )
9 df-disj 3823 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x  e.  A  z  e.  B )
10 df-disj 3823 . 2  |-  (Disj  y  e.  A  C  <->  A. z E* y  e.  A  z  e.  C )
118, 9, 103bitr4i 210 1  |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   F/_wnfc 2215   E*wrmo 2362  Disj wdisj 3822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-reu 2366  df-rmo 2367  df-disj 3823
This theorem is referenced by:  cbvdisjv  3833  disjnims  3837
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