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Theorem cbvdisj 3987
Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
cbvdisj.1 𝑦𝐵
cbvdisj.2 𝑥𝐶
cbvdisj.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvdisj (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbvdisj
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvdisj.1 . . . . 5 𝑦𝐵
21nfcri 2313 . . . 4 𝑦 𝑧𝐵
3 cbvdisj.2 . . . . 5 𝑥𝐶
43nfcri 2313 . . . 4 𝑥 𝑧𝐶
5 cbvdisj.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2247 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrmo 2702 . . 3 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐴 𝑧𝐶)
87albii 1470 . 2 (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
9 df-disj 3978 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
10 df-disj 3978 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
118, 9, 103bitr4i 212 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351   = wceq 1353  wcel 2148  wnfc 2306  ∃*wrmo 2458  Disj wdisj 3977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-reu 2462  df-rmo 2463  df-disj 3978
This theorem is referenced by:  cbvdisjv  3988  disjnims  3992
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