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Theorem disjeq2 4552
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjeq2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))

Proof of Theorem disjeq2
StepHypRef Expression
1 eqimss2 3621 . . . 4 (𝐵 = 𝐶𝐶𝐵)
21ralimi 2936 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐶𝐵)
3 disjss2 4551 . . 3 (∀𝑥𝐴 𝐶𝐵 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
42, 3syl 17 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
5 eqimss 3620 . . . 4 (𝐵 = 𝐶𝐵𝐶)
65ralimi 2936 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐵𝐶)
7 disjss2 4551 . . 3 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
86, 7syl 17 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
94, 8impbid 201 1 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wral 2896  wss 3540  Disj wdisj 4548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-rmo 2904  df-in 3547  df-ss 3554  df-disj 4549
This theorem is referenced by:  disjeq2dv  4553  voliun  23074  carsgclctunlem2  29502  mblfinlem2  32411  voliunnfl  32417
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