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Theorem cbvdisj 3974
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 2306 . . . 4  |-  F/ y  z  e.  B
3 cbvdisj.2 . . . . 5  |-  F/_ x C
43nfcri 2306 . . . 4  |-  F/ x  z  e.  C
5 cbvdisj.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2240 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvrmo 2695 . . 3  |-  ( E* x  e.  A  z  e.  B  <->  E* y  e.  A  z  e.  C )
87albii 1463 . 2  |-  ( A. z E* x  e.  A  z  e.  B  <->  A. z E* y  e.  A  z  e.  C )
9 df-disj 3965 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x  e.  A  z  e.  B )
10 df-disj 3965 . 2  |-  (Disj  y  e.  A  C  <->  A. z E* y  e.  A  z  e.  C )
118, 9, 103bitr4i 211 1  |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1346    = wceq 1348    e. wcel 2141   F/_wnfc 2299   E*wrmo 2451  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-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-reu 2455  df-rmo 2456  df-disj 3965
This theorem is referenced by:  cbvdisjv  3975  disjnims  3979
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