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Theorem cbvdisj 4003
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 2413 . . . 4  |-  F/ y  z  e.  B
3 cbvdisj.2 . . . . 5  |-  F/_ x C
43nfcri 2413 . . . 4  |-  F/ x  z  e.  C
5 cbvdisj.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2350 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvrmo 2763 . . 3  |-  ( E* x  e.  A z  e.  B  <->  E* y  e.  A z  e.  C
)
87albii 1553 . 2  |-  ( A. z E* x  e.  A
z  e.  B  <->  A. z E* y  e.  A
z  e.  C )
9 df-disj 3994 . 2  |-  (Disj  x  e.  A B  <->  A. z E* x  e.  A
z  e.  B )
10 df-disj 3994 . 2  |-  (Disj  y  e.  A C  <->  A. z E* y  e.  A
z  e.  C )
118, 9, 103bitr4i 268 1  |-  (Disj  x  e.  A B  <-> Disj  y  e.  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   F/_wnfc 2406   E*wrmo 2546  Disj wdisj 3993
This theorem is referenced by:  cbvdisjv  4004  disjors  4009  disjxiun  4020  volfiniun  18904  voliun  18911  hashunif  23385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-disj 3994
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