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Theorem cbvdisj 3952
 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 2293 . . . 4 𝑦 𝑧𝐵
3 cbvdisj.2 . . . . 5 𝑥𝐶
43nfcri 2293 . . . 4 𝑥 𝑧𝐶
5 cbvdisj.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2227 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrmo 2679 . . 3 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐴 𝑧𝐶)
87albii 1450 . 2 (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
9 df-disj 3943 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
10 df-disj 3943 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
118, 9, 103bitr4i 211 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333   = wceq 1335   ∈ wcel 2128  Ⅎwnfc 2286  ∃*wrmo 2438  Disj wdisj 3942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-reu 2442  df-rmo 2443  df-disj 3943 This theorem is referenced by:  cbvdisjv  3953  disjnims  3957
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