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Theorem cbvdisjv 5038
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 2981 . 2 𝑦𝐵
2 nfcv 2981 . 2 𝑥𝐶
3 cbvdisjv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvdisj 5037 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  Disj wdisj 5027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-disj 5028
This theorem is referenced by:  uniioombllem4  24102  hashunif  30442  tocyccntz  30701  totprob  31572  disjrnmpt2  41311  ismeannd  42612  psmeasure  42616  volmea  42619  meaiuninclem  42625  caratheodorylem1  42671  caratheodory  42673
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