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Theorem 3eltr3i 2902
 Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3i.1 𝐴𝐵
3eltr3i.2 𝐴 = 𝐶
3eltr3i.3 𝐵 = 𝐷
Assertion
Ref Expression
3eltr3i 𝐶𝐷

Proof of Theorem 3eltr3i
StepHypRef Expression
1 3eltr3i.2 . 2 𝐴 = 𝐶
2 3eltr3i.1 . . 3 𝐴𝐵
3 3eltr3i.3 . . 3 𝐵 = 𝐷
42, 3eleqtri 2888 . 2 𝐴𝐷
51, 4eqeltrri 2887 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:  raddcn  31297  clsk1independent  40792  fourierdlem62  42853
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