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Theorem cbvdisjv 5120
Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
cbvdisjv.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvdisjv (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvdisjv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvdisjv.1 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
21eleq2d 2826 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
32cbvrmovw 3402 . . 3 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐴 𝑧𝐶)
43albii 1818 . 2 (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
5 df-disj 5110 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
6 df-disj 5110 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
74, 5, 63bitr4i 303 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1537   = wceq 1539  wcel 2107  ∃*wrmo 3378  Disj wdisj 5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2539  df-cleq 2728  df-clel 2815  df-rmo 3379  df-disj 5110
This theorem is referenced by:  uniioombllem4  25622  hashunif  32811  tocyccntz  33165  totprob  34430  disjrnmpt2  45198  ismeannd  46487  psmeasure  46491  volmea  46494  meaiuninclem  46500  caratheodorylem1  46546  caratheodory  46548
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