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Theorem 3eltr3d 2120
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3d.1  |-  ( ph  ->  A  e.  B )
3eltr3d.2  |-  ( ph  ->  A  =  C )
3eltr3d.3  |-  ( ph  ->  B  =  D )
Assertion
Ref Expression
3eltr3d  |-  ( ph  ->  C  e.  D )

Proof of Theorem 3eltr3d
StepHypRef Expression
1 3eltr3d.2 . 2  |-  ( ph  ->  A  =  C )
2 3eltr3d.1 . . 3  |-  ( ph  ->  A  e.  B )
3 3eltr3d.3 . . 3  |-  ( ph  ->  B  =  D )
42, 3eleqtrd 2116 . 2  |-  ( ph  ->  A  e.  D )
51, 4eqeltrrd 2115 1  |-  ( ph  ->  C  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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:  reg3exmidlemwe  4303  nnaordi  6081  icoshftf1o  8859  lincmb01cmp  8871  fzosubel  9050
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