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Theorem 3eltr4g 2521
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 2503 . 2  |-  ( C  e.  D  <->  A  e.  B )
51, 4sylibr 205 1  |-  ( ph  ->  C  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726
This theorem is referenced by:  rankelun  7800  rankelpr  7801  rankelop  7802  cdivcncf  18949  itg1addlem4  19593  cxpcn3  20634  bposlem4  21073  mapfzcons  26774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2431  df-clel 2434
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