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Theorem disjeq2 5118
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 4042 . . . 4 (𝐵 = 𝐶𝐶𝐵)
21ralimi 3084 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐶𝐵)
3 disjss2 5117 . . 3 (∀𝑥𝐴 𝐶𝐵 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
42, 3syl 17 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
5 eqimss 4041 . . . 4 (𝐵 = 𝐶𝐵𝐶)
65ralimi 3084 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐵𝐶)
7 disjss2 5117 . . 3 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
86, 7syl 17 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
94, 8impbid 211 1 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wral 3062  wss 3949  Disj wdisj 5114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rmo 3377  df-v 3477  df-in 3956  df-ss 3966  df-disj 5115
This theorem is referenced by:  disjeq2dv  5119  voliun  25071  carsgclctunlem2  33318  mblfinlem2  36526  voliunnfl  36532
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