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Theorem cbvdisj 3755
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 2172 . . . 4  |-  F/ y  z  e.  B
3 cbvdisj.2 . . . . 5  |-  F/_ x C
43nfcri 2172 . . . 4  |-  F/ x  z  e.  C
5 cbvdisj.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2107 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvrmo 2532 . . 3  |-  ( E* x  e.  A  z  e.  B  <->  E* y  e.  A  z  e.  C )
87albii 1359 . 2  |-  ( A. z E* x  e.  A  z  e.  B  <->  A. z E* y  e.  A  z  e.  C )
9 df-disj 3746 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x  e.  A  z  e.  B )
10 df-disj 3746 . 2  |-  (Disj  y  e.  A  C  <->  A. z E* y  e.  A  z  e.  C )
118, 9, 103bitr4i 201 1  |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   A.wal 1241    = wceq 1243    e. wcel 1393   F/_wnfc 2165   E*wrmo 2309  Disj wdisj 3745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-reu 2313  df-rmo 2314  df-disj 3746
This theorem is referenced by:  cbvdisjv  3756
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