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Theorem 3eltr4i 2872
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 2858 . 2 𝐴𝐷
51, 4eqeltri 2855 1 𝐶𝐷
Colors of variables: wff setvar class
Syntax hints:   = wceq 1508  wcel 2051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-cleq 2764  df-clel 2839
This theorem is referenced by:  oancom  8906  0r  10298  1sr  10299  m1r  10300  recvs  23468  qcvs  23469  wlk2v2elem1  27699  konigsbergiedgw  27795  lmxrge0  30871  brsigarn  31120  sinccvglem  32472  bj-minftyccb  34013  fouriersw  41981
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