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Theorem cbvdisj 4764
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 2907 . . . 4 𝑦 𝑧𝐵
3 cbvdisj.2 . . . . 5 𝑥𝐶
43nfcri 2907 . . . 4 𝑥 𝑧𝐶
5 cbvdisj.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2836 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrmo 3319 . . 3 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐴 𝑧𝐶)
87albii 1895 . 2 (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
9 df-disj 4755 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
10 df-disj 4755 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
118, 9, 103bitr4i 292 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629   = wceq 1631  wcel 2145  wnfc 2900  ∃*wrmo 3064  Disj wdisj 4754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-disj 4755
This theorem is referenced by:  cbvdisjv  4765  disjors  4769  disjxiun  4783  volfiniun  23535  voliun  23542  carsggect  30720  omsmeas  30725  disjf1  39889  disjrnmpt2  39895  fsumiunss  40325  sge0iunmpt  41152  iundjiun  41194  meadjiun  41200
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