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Theorem disjeq1 3752
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjeq1 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem disjeq1
StepHypRef Expression
1 eqimss2 2998 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 disjss1 3751 . . 3 (𝐵𝐴 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
31, 2syl 14 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
4 eqimss 2997 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 disjss1 3751 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
64, 5syl 14 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
73, 6impbid 120 1 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wss 2917  Disj wdisj 3745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-rmo 2314  df-in 2924  df-ss 2931  df-disj 3746
This theorem is referenced by:  disjeq1d  3753
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