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Theorem cbvdisjv 3785
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 2220 . 2  |-  F/_ y B
2 nfcv 2220 . 2  |-  F/_ x C
3 cbvdisjv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbvdisj 3784 1  |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285  Disj wdisj 3774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-reu 2356  df-rmo 2357  df-disj 3775
This theorem is referenced by: (None)
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