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Theorem cbvdisj 4192
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 2566 . . . 4  |-  F/ y  z  e.  B
3 cbvdisj.2 . . . . 5  |-  F/_ x C
43nfcri 2566 . . . 4  |-  F/ x  z  e.  C
5 cbvdisj.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2503 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvrmo 2931 . . 3  |-  ( E* x  e.  A z  e.  B  <->  E* y  e.  A z  e.  C
)
87albii 1575 . 2  |-  ( A. z E* x  e.  A
z  e.  B  <->  A. z E* y  e.  A
z  e.  C )
9 df-disj 4183 . 2  |-  (Disj  x  e.  A B  <->  A. z E* x  e.  A
z  e.  B )
10 df-disj 4183 . 2  |-  (Disj  y  e.  A C  <->  A. z E* y  e.  A
z  e.  C )
118, 9, 103bitr4i 269 1  |-  (Disj  x  e.  A B  <-> Disj  y  e.  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    = wceq 1652    e. wcel 1725   F/_wnfc 2559   E*wrmo 2708  Disj wdisj 4182
This theorem is referenced by:  cbvdisjv  4193  disjors  4198  disjxiun  4209  volfiniun  19441  voliun  19448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-disj 4183
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