Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  disjeq1 Structured version   GIF version

Theorem disjeq1 3743
 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 2992 . . 3 (A = BBA)
2 disjss1 3742 . . 3 (BA → (Disj x A 𝐶Disj x B 𝐶))
31, 2syl 14 . 2 (A = B → (Disj x A 𝐶Disj x B 𝐶))
4 eqimss 2991 . . 3 (A = BAB)
5 disjss1 3742 . . 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 1242   ⊆ wss 2911  Disj wdisj 3736 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-rmo 2308  df-in 2918  df-ss 2925  df-disj 3737 This theorem is referenced by:  disjeq1d  3744
 Copyright terms: Public domain W3C validator