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Theorem disjeq1 3726
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjeq1 (A = B → (Disj x A 𝐶Disj x B 𝐶))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝐶(x)

Proof of Theorem disjeq1
StepHypRef Expression
1 eqimss2 2975 . . 3 (A = BBA)
2 disjss1 3725 . . 3 (BA → (Disj x A 𝐶Disj x B 𝐶))
31, 2syl 14 . 2 (A = B → (Disj x A 𝐶Disj x B 𝐶))
4 eqimss 2974 . . 3 (A = BAB)
5 disjss1 3725 . . 3 (AB → (Disj x B 𝐶Disj x A 𝐶))
64, 5syl 14 . 2 (A = B → (Disj x B 𝐶Disj x A 𝐶))
73, 6impbid 120 1 (A = B → (Disj x A 𝐶Disj x B 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228  wss 2894  Disj wdisj 3719
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-rmo 2292  df-in 2901  df-ss 2908  df-disj 3720
This theorem is referenced by:  disjeq1d  3727
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