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Theorem 3eltr4i 2854
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 2840 . 2 𝐴𝐷
51, 4eqeltri 2837 1 𝐶𝐷
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-clel 2816
This theorem is referenced by:  oancom  9691  0r  11120  1sr  11121  m1r  11122  smndex1ibas  18913  recvs  25179  recvsOLD  25180  qcvs  25181  wlk2v2elem1  30174  konigsbergiedgw  30267  lmxrge0  33951  brsigarn  34185  ex-sategoelel12  35432  sinccvglem  35677  bj-minftyccb  37226  resuppsinopn  42393  omcl3g  43347  fouriersw  46246
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