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Theorem cbvdisjv 4193
Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
cbvdisjv.1  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbvdisjv  |-  (Disj  x  e.  A B  <-> Disj  y  e.  A C )
Distinct variable groups:    x, y, A    y, B    x, C
Allowed substitution hints:    B( x)    C( y)

Proof of Theorem cbvdisjv
StepHypRef Expression
1 nfcv 2572 . 2  |-  F/_ y B
2 nfcv 2572 . 2  |-  F/_ x C
3 cbvdisjv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbvdisj 4192 1  |-  (Disj  x  e.  A B  <-> Disj  y  e.  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652  Disj wdisj 4182
This theorem is referenced by:  uniioombllem4  19478  hashunif  24158  totprob  24685
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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