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Theorem cbvdisj 3755
 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 2172 . . . 4 𝑦 𝑧𝐵
3 cbvdisj.2 . . . . 5 𝑥𝐶
43nfcri 2172 . . . 4 𝑥 𝑧𝐶
5 cbvdisj.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2107 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrmo 2532 . . 3 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐴 𝑧𝐶)
87albii 1359 . 2 (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
9 df-disj 3746 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
10 df-disj 3746 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
118, 9, 103bitr4i 201 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Ⅎwnfc 2165  ∃*wrmo 2309  Disj wdisj 3745 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-reu 2313  df-rmo 2314  df-disj 3746 This theorem is referenced by:  cbvdisjv  3756
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