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Theorem disjeq2 5032
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 4028 . . . 4 (𝐵 = 𝐶𝐶𝐵)
21ralimi 3165 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐶𝐵)
3 disjss2 5031 . . 3 (∀𝑥𝐴 𝐶𝐵 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
42, 3syl 17 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
5 eqimss 4027 . . . 4 (𝐵 = 𝐶𝐵𝐶)
65ralimi 3165 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐵𝐶)
7 disjss2 5031 . . 3 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
86, 7syl 17 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
94, 8impbid 213 1 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wral 3143  wss 3940  Disj wdisj 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-clab 2805  df-cleq 2819  df-clel 2898  df-ral 3148  df-rmo 3151  df-in 3947  df-ss 3956  df-disj 5029
This theorem is referenced by:  disjeq2dv  5033  voliun  24070  carsgclctunlem2  31463  mblfinlem2  34797  voliunnfl  34803
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