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Theorem disjeq2 4756
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 3805 . . . 4 (𝐵 = 𝐶𝐶𝐵)
21ralimi 3100 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐶𝐵)
3 disjss2 4755 . . 3 (∀𝑥𝐴 𝐶𝐵 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
42, 3syl 17 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
5 eqimss 3804 . . . 4 (𝐵 = 𝐶𝐵𝐶)
65ralimi 3100 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐵𝐶)
7 disjss2 4755 . . 3 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
86, 7syl 17 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
94, 8impbid 202 1 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1630  wral 3060  wss 3721  Disj wdisj 4752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-ral 3065  df-rmo 3068  df-in 3728  df-ss 3735  df-disj 4753
This theorem is referenced by:  disjeq2dv  4757  voliun  23541  carsgclctunlem2  30715  mblfinlem2  33773  voliunnfl  33779
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