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Theorem disjeq2 3918
 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 3157 . . . 4 (𝐵 = 𝐶𝐶𝐵)
21ralimi 2498 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐶𝐵)
3 disjss2 3917 . . 3 (∀𝑥𝐴 𝐶𝐵 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
42, 3syl 14 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
5 eqimss 3156 . . . 4 (𝐵 = 𝐶𝐵𝐶)
65ralimi 2498 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 𝐵𝐶)
7 disjss2 3917 . . 3 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
86, 7syl 14 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
94, 8impbid 128 1 (∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332  ∀wral 2417   ⊆ wss 3076  Disj wdisj 3914 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-rmo 2425  df-in 3082  df-ss 3089  df-disj 3915 This theorem is referenced by:  disjeq2dv  3919
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