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Theorem 3eltr3g 2172
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3g.1  |-  ( ph  ->  A  e.  B )
3eltr3g.2  |-  A  =  C
3eltr3g.3  |-  B  =  D
Assertion
Ref Expression
3eltr3g  |-  ( ph  ->  C  e.  D )

Proof of Theorem 3eltr3g
StepHypRef Expression
1 3eltr3g.1 . 2  |-  ( ph  ->  A  e.  B )
2 3eltr3g.2 . . 3  |-  A  =  C
3 3eltr3g.3 . . 3  |-  B  =  D
42, 3eleq12i 2155 . 2  |-  ( A  e.  B  <->  C  e.  D )
51, 4sylib 120 1  |-  ( ph  ->  C  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084
This theorem is referenced by: (None)
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