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Theorem cbvdisj 4557
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 2744 . . . 4 𝑦 𝑧𝐵
3 cbvdisj.2 . . . . 5 𝑥𝐶
43nfcri 2744 . . . 4 𝑥 𝑧𝐶
5 cbvdisj.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2672 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrmo 3145 . . 3 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐴 𝑧𝐶)
87albii 1736 . 2 (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
9 df-disj 4548 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
10 df-disj 4548 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
118, 9, 103bitr4i 290 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472   = wceq 1474  wcel 1976  wnfc 2737  ∃*wrmo 2898  Disj wdisj 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-disj 4548
This theorem is referenced by:  cbvdisjv  4558  disjors  4562  disjxiun  4573  disjxiunOLD  4574  volfiniun  23039  voliun  23046  carsggect  29513  omsmeas  29518  disjf1  38160  disjrnmpt2  38166  fsumiunss  38439  sge0iunmpt  39108  iundjiun  39150  meadjiun  39156
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