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Theorem 3eltr3i 2118
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3.1  |-  A  e.  B
3eltr3.2  |-  A  =  C
3eltr3.3  |-  B  =  D
Assertion
Ref Expression
3eltr3i  |-  C  e.  D

Proof of Theorem 3eltr3i
StepHypRef Expression
1 3eltr3.2 . 2  |-  A  =  C
2 3eltr3.1 . . 3  |-  A  e.  B
3 3eltr3.3 . . 3  |-  B  =  D
42, 3eleqtri 2112 . 2  |-  A  e.  D
51, 4eqeltrri 2111 1  |-  C  e.  D
Colors of variables: wff set class
Syntax hints:    = wceq 1243    e. wcel 1393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036
This theorem is referenced by: (None)
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