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Theorem cbvdisjv 4004
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 2419 . 2  |-  F/_ y B
2 nfcv 2419 . 2  |-  F/_ x C
3 cbvdisjv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbvdisj 4003 1  |-  (Disj  x  e.  A B  <-> Disj  y  e.  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623  Disj wdisj 3993
This theorem is referenced by:  uniioombllem4  18941  totprob  23630
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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