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Theorem cbvdisjv 4826
Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
cbvdisjv.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvdisjv (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 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 2945 . 2 𝑦𝐵
2 nfcv 2945 . 2 𝑥𝐶
3 cbvdisjv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvdisj 4825 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  Disj wdisj 4815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ral 3098  df-rex 3099  df-reu 3100  df-rmo 3101  df-disj 4816
This theorem is referenced by:  uniioombllem4  23698  hashunif  30084  totprob  31010  disjrnmpt2  40133  ismeannd  41431  psmeasure  41435  volmea  41438  meaiuninclem  41444  caratheodorylem1  41490  caratheodory  41492
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