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Theorem 3eltr4i 2903
 Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr4i.1 𝐴𝐵
3eltr4i.2 𝐶 = 𝐴
3eltr4i.3 𝐷 = 𝐵
Assertion
Ref Expression
3eltr4i 𝐶𝐷

Proof of Theorem 3eltr4i
StepHypRef Expression
1 3eltr4i.2 . 2 𝐶 = 𝐴
2 3eltr4i.1 . . 3 𝐴𝐵
3 3eltr4i.3 . . 3 𝐷 = 𝐵
42, 3eleqtrri 2889 . 2 𝐴𝐷
51, 4eqeltri 2886 1 𝐶𝐷
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111 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 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870 This theorem is referenced by:  oancom  9101  0r  10494  1sr  10495  m1r  10496  smndex1ibas  18060  recvs  23761  qcvs  23762  wlk2v2elem1  27950  konigsbergiedgw  28043  lmxrge0  31320  brsigarn  31568  ex-sategoelel12  32802  sinccvglem  33043  bj-minftyccb  34659  fouriersw  42916
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