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Theorem cbvdisj 3746
 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 x A BDisj y A 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 2169 . . . 4 y z B
3 cbvdisj.2 . . . . 5 x𝐶
43nfcri 2169 . . . 4 x z 𝐶
5 cbvdisj.3 . . . . 5 (x = yB = 𝐶)
65eleq2d 2104 . . . 4 (x = y → (z Bz 𝐶))
72, 4, 6cbvrmo 2526 . . 3 (∃*x A z B∃*y A z 𝐶)
87albii 1356 . 2 (z∃*x A z Bz∃*y A z 𝐶)
9 df-disj 3737 . 2 (Disj x A Bz∃*x A z B)
10 df-disj 3737 . 2 (Disj y A 𝐶z∃*y A z 𝐶)
118, 9, 103bitr4i 201 1 (Disj x A BDisj y A 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Ⅎwnfc 2162  ∃*wrmo 2303  Disj wdisj 3736 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-reu 2307  df-rmo 2308  df-disj 3737 This theorem is referenced by:  cbvdisjv  3747
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