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Theorem disjeq1 3916
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 3152 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 disjss1 3915 . . 3 (𝐵𝐴 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
31, 2syl 14 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
4 eqimss 3151 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 disjss1 3915 . . 3 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
64, 5syl 14 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
73, 6impbid 128 1 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wss 3071  Disj wdisj 3909
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-rmo 2424  df-in 3077  df-ss 3084  df-disj 3910
This theorem is referenced by:  disjeq1d  3917
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