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Theorem 3eltr4g 2203
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr4g.1  |-  ( ph  ->  A  e.  B )
3eltr4g.2  |-  C  =  A
3eltr4g.3  |-  D  =  B
Assertion
Ref Expression
3eltr4g  |-  ( ph  ->  C  e.  D )

Proof of Theorem 3eltr4g
StepHypRef Expression
1 3eltr4g.1 . 2  |-  ( ph  ->  A  e.  B )
2 3eltr4g.2 . . 3  |-  C  =  A
3 3eltr4g.3 . . 3  |-  D  =  B
42, 3eleq12i 2185 . 2  |-  ( C  e.  D  <->  A  e.  B )
51, 4sylibr 133 1  |-  ( ph  ->  C  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113
This theorem is referenced by:  riotacl2  5711  2strop1g  11991
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