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Theorem 3eltr3i 2160
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3.1 𝐴𝐵
3eltr3.2 𝐴 = 𝐶
3eltr3.3 𝐵 = 𝐷
Assertion
Ref Expression
3eltr3i 𝐶𝐷

Proof of Theorem 3eltr3i
StepHypRef Expression
1 3eltr3.2 . 2 𝐴 = 𝐶
2 3eltr3.1 . . 3 𝐴𝐵
3 3eltr3.3 . . 3 𝐵 = 𝐷
42, 3eleqtri 2154 . 2 𝐴𝐷
51, 4eqeltrri 2153 1 𝐶𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078
This theorem is referenced by: (None)
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