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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 1601  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-cleq 2770  df-clel 2774
This theorem is referenced by:  oancom  8845  0r  10237  1sr  10238  m1r  10239  recvs  23353  qcvs  23354  wlk2v2elem1  27558  konigsbergiedgw  27654  lmxrge0  30596  brsigarn  30845  sinccvglem  32163  bj-minftyccb  33702  fouriersw  41375
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