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Theorem 3eltr3i 2374
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 2368 . 2  |-  A  e.  D
51, 4eqeltrri 2367 1  |-  C  e.  D
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696
This theorem is referenced by:  raddcn  23317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-cleq 2289  df-clel 2292
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