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Theorem 3eltr4i 2221
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr4.1  |-  A  e.  B
3eltr4.2  |-  C  =  A
3eltr4.3  |-  D  =  B
Assertion
Ref Expression
3eltr4i  |-  C  e.  D

Proof of Theorem 3eltr4i
StepHypRef Expression
1 3eltr4.2 . 2  |-  C  =  A
2 3eltr4.1 . . 3  |-  A  e.  B
3 3eltr4.3 . . 3  |-  D  =  B
42, 3eleqtrri 2215 . 2  |-  A  e.  D
51, 4eqeltri 2212 1  |-  C  e.  D
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135
This theorem is referenced by:  1nq  7186  0r  7570  1sr  7571  m1r  7572
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