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Theorem disjeq1f 30339
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disjss1f.1 𝑥𝐴
disjss1f.2 𝑥𝐵
Assertion
Ref Expression
disjeq1f (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))

Proof of Theorem disjeq1f
StepHypRef Expression
1 eqimss2 3975 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 disjss1f.2 . . . 4 𝑥𝐵
3 disjss1f.1 . . . 4 𝑥𝐴
42, 3disjss1f 30338 . . 3 (𝐵𝐴 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
51, 4syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
6 eqimss 3974 . . 3 (𝐴 = 𝐵𝐴𝐵)
73, 2disjss1f 30338 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
86, 7syl 17 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
95, 8impbid 215 1 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wnfc 2939  wss 3884  Disj wdisj 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rmo 3117  df-v 3446  df-in 3891  df-ss 3901  df-disj 4999
This theorem is referenced by:  ldgenpisyslem1  31530
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