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Theorem 3eltr3g 2365
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 2348 . 2  |-  ( A  e.  B  <->  C  e.  D )
51, 4sylib 188 1  |-  ( ph  ->  C  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684
This theorem is referenced by:  rmulccn  23301  esumpfinvallem  23442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-cleq 2276  df-clel 2279
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